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Learning Objectives
- Use the properties of angles
- Use the properties of triangles
- Use the Pythagorean Theorem
be prepared!
Before you get started, take this readiness quiz.
- Solve: x + 3 + 6 = 11. If you missed this problem, review Example 8.1.6.
- Solve: \(\dfrac{a}{45} = \dfrac{4}{3}\). If you missed this problem, review Example 6.5.3.
- Simplify: \(\sqrt{36 + 64}\). If you missed this problem, review Example 5.12.4.
So far in this chapter, we have focused on solving word problems, which are similar to many real-world applications of algebra. In the next few sections, we will apply our problem-solving strategies to some common geometry problems.
Use the Properties of Angles
Are you familiar with the phrase ‘do a 180’? It means to turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is 180 degrees. See Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\)
An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In Figure \(\PageIndex{2}\), ∠A is the angle with vertex at point A. The measure of ∠A is written m ∠ A.
Figure \(\PageIndex{2}\) - ∠ A is the angle with vertex at point A.
We measure angles in degrees, and use the symbol ° to represent degrees. We use the abbreviation m for the measure of an angle. So if ∠A is 27°, we would write m ∠ A = 27.
If the sum of the measures of two angles is 180°, then they are called supplementary angles. In Figure \(\PageIndex{3}\), each pair of angles is supplementary because their measures add to 180°. Each angle is the supplement of the other.
Figure \(\PageIndex{3}\) - The sum of the measures of supplementary angles is 180°.
If the sum of the measures of two angles is 90°, then the angles are complementary angles. In Figure \(\PageIndex{4}\), each pair of angles is complementary, because their measures add to 90°. Each angle is the complement of the other.
Figure \(\PageIndex{4}\) - The sum of the measures of complementary angles is 90°.
Definition: Supplementary and Complementary Angles
If the sum of the measures of two angles is 180°, then the angles are supplementary.
If ∠A and ∠B are supplementary, then m∠A + m∠B = 180°.
If the sum of the measures of two angles is 90°, then the angles are complementary.
If ∠A and ∠B are complementary, then m∠A + m∠B = 90°.
In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.
In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.
HOW TO: USE A PROBLEM SOLVING STRATEGY FOR GEOMETRY APPLICATIONS
Step 1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
Step 2. Identify what you are looking for
Step 3. Name what you are looking for and choose a variable to represent it.
Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.
The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.
Example \(\PageIndex{1}\):
An angle measures 40°. Find (a) its supplement, and (b) its complement.
Solution
(a)
| Step 1. Read the problem. Draw the figure and label it with the given information. |  |
| Step 2. Identify what you are looking for. | the supplement of a 40° |
| Step 3. Name. Choose a variable to represent it. | let s = the measure of the supplement |
| Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. | $$m \angle A + m \angle B = 180$$ |
| Step 5. Solve the equation. | $$\begin{split} s + 40 &= 180 \\ s &= 140 \end{split}$$ |
| Step 6. Check. | $$\begin{split} 140 + 40 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$ |
| Step 7. Answer the question. | The supplement of the 40° angle is 140°. |
(b)
| Step 1. Read the problem. Draw the figure and label it with the given information. |  |
| Step 2. Identify what you are looking for. | the complement of a 40° |
| Step 3. Name. Choose a variable to represent it. | let c = the measure of the complement |
| Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. | $$m \angle A + m \angle B = 90$$ |
| Step 5. Solve the equation. | $$\begin{split} c + 40 &= 90 \\ c &= 50 \end{split}$$ |
| Step 6. Check. | $$\begin{split} 50 + 40 &\stackrel{?}{=} 90 \\ 90 &= 90\; \checkmark \end{split}$$ |
| Step 7. Answer the question. | The supplement of the 40° angle is 50°. |
Exercise \(\PageIndex{1}\):
An angle measures 25°. Find (a) its supplement, and (b) its complement.
- Answer a
-
155°
- Answer b
-
65°
Exercise \(\PageIndex{2}\):
An angle measures 77°. Find (a) its supplement, and (b) its complement.
- Answer a
-
103°
- Answer b
-
13°
Did you notice that the words complementary and supplementary are in alphabetical order just like 90 and 180 are in numerical order?
Example \(\PageIndex{2}\):
Two angles are supplementary. The larger angle is 30° more than the smaller angle. Find the measure of both angles.
Solution
| Step 1. Read the problem. Draw the figure and label it with the given information. |  |
| Step 2. Identify what you are looking for. | the measures of both angles |
| Step 3. Name. Choose a variable to represent it. | let a = measure of the smaller angle a + 30 = measure of larger angle |
| Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. | $$m \angle A + m \angle B = 180$$ |
| Step 5. Solve the equation. | $$\begin{split} (a + 30) + a &= 180 \\ 2a + 30 &= 180 \\ 2a &= 150 \\ a &= 75\quad measure\; of\; smaller\; angle \\ a &+ 30\quad measure\; of\; larger\; angle \\ 75 &+ 30 \\ &105 \end{split}$$ |
| Step 6. Check. | $$\begin{split} m \angle A + m \angle B &= 180 \\ 75 + 105 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$ |
| Step 7. Answer the question. | The measures of the angles are 75° and 105°. |
Exercise \(\PageIndex{3}\):
Two angles are supplementary. The larger angle is 100° more than the smaller angle. Find the measures of both angles.
- Answer
-
40°, 140°
Exercise \(\PageIndex{4}\):
Two angles are complementary. The larger angle is 40° more than the smaller angle. Find the measures of both angles.
- Answer
-
25°, 65°
Use the Properties of Triangles
What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in Figure \(\PageIndex{5}\) is called ΔABC, read ‘triangle ABC’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.
Figure \(\PageIndex{5}\) - ΔABC has vertices A, B, and C and sides a, b, and c.
The three angles of a triangle are related in a special way. The sum of their measures is 180°.
\[m \angle A + m \angle B + m \angle C = 180°\]
Definition: Sum of the Measures of the Angles of a Triangle
For any ΔABC, the sum of the measures of the angles is 180°.
\[m \angle A + m \angle B + m \angle C = 180°\]
Example \(\PageIndex{3}\):
The measures of two angles of a triangle are 55° and 82°. Find the measure of the third angle.
Solution
| Step 1. Read the problem. Draw the figure and label it with the given information. |  |
| Step 2. Identify what you are looking for. | the measure of the third angle in a triangle |
| Step 3. Name. Choose a variable to represent it. | let x = the measure of the angle |
| Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. | $$m \angle A + m \angle B + m \angle C = 180$$ |
| Step 5. Solve the equation. | $$\begin{split} 55 + 82 + x &= 180 \\ 137 + x &= 180 \\ x &= 43 \end{split}$$ |
| Step 6. Check. | $$\begin{split} 55 + 82 + 43 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$ |
| Step 7. Answer the question. | The measure of the third angle is 43 degrees. |
Exercise \(\PageIndex{5}\):
The measures of two angles of a triangle are 31° and 128°. Find the measure of the third angle.
- Answer
-
21°
Exercise \(\PageIndex{6}\):
A triangle has angles of 49° and 75°. Find the measure of the third angle.
- Answer
-
56°
(Video) Pythagorean Theorem | MathHelp.com
Right Triangles
Some triangles have special names. We will look first at the right triangle. A right triangle has one 90° angle, which is often marked with the symbol shown in Figure \(\PageIndex{6}\).
Figure \(\PageIndex{6}\)
If we know that a triangle is a right triangle, we know that one angle measures 90° so we only need the measure of one of the other angles in order to determine the measure of the third angle.
Example \(\PageIndex{4}\):
One angle of a right triangle measures 28°. What is the measure of the third angle?
Solution
| Step 1. Read the problem. Draw the figure and label it with the given information. |  |
| Step 2. Identify what you are looking for. | the measure of an angle |
| Step 3. Name. Choose a variable to represent it. | let x = the measure of the angle |
| Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. | $$m \angle A + m \angle B + m \angle C = 180$$ |
| Step 5. Solve the equation. | $$\begin{split} x + 90 + 28 &= 180 \\ x + 118 &= 180 \\ x &= 62 \end{split}$$ |
| Step 6. Check. | $$\begin{split} 180 &\stackrel{?}{=} 90 + 28 + 62 \\ 180 &= 180\; \checkmark \end{split}$$ |
| Step 7. Answer the question. | The measure of the third angle is 62°. |
Exercise \(\PageIndex{7}\):
One angle of a right triangle measures 56°. What is the measure of the other angle?
- Answer
-
34°
Exercise \(\PageIndex{8}\):
One angle of a right triangle measures 45°. What is the measure of the other angle?
- Answer
-
45°
In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.
Example \(\PageIndex{5}\):
The measure of one angle of a right triangle is 20° more than the measure of the smallest angle. Find the measures of all three angles.
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what you are looking for. | the measures of all three angles |
| Step 3. Name. Choose a variable to represent it. Now draw the figure and label it with the given information. | Let a = 1st angle a + 20 = 2nd angle 90 = 3rd angle (the right angle)
|
| Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. | $$\begin{split} m \angle A + m \angle B + m \angle C &= 180 \\ a + (a + 20) + 90 &= 180 \end{split}$$ |
| Step 5. Solve the equation. | $$\begin{split} 2a + 110 &= 180 \\ 2a &= 70 \\ a &= 35 \quad first\; angle \\ a + &20 \quad second\; angle \\ \textcolor{red}{35} + &20 \\ &55 \\ &90 \quad third\; angle \end{split}$$ |
| Step 6. Check. | $$\begin{split} 35 + 55 + 90 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$ |
| Step 7. Answer the question. | The three angles measure 35°, 55°, and 90°. |
Exercise \(\PageIndex{9}\):
The measure of one angle of a right triangle is 50° more than the measure of the smallest angle. Find the measures of all three angles.
- Answer
-
20°, 70°, 90°
Exercise \(\PageIndex{10}\):
The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles.
- Answer
-
30°, 60°, 90°
(Video) ABE 84 9.3 Use Properties of Angles, Triangles and Pythagorean Theorem
Similar Triangles
When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.
The two triangles in Figure \(\PageIndex{7}\) are similar. Each side of ΔABC is four times the length of the corresponding side of ΔXYZ and their corresponding angles have equal measures.
Figure \(\PageIndex{7}\) - ΔABC and ΔXYZ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.
Definition: Properties of Similar Triangles
If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.
The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in ΔABC:
the length a can also be written BC
the length b can also be written AC
the length c can also be written AB
We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.
Example \(\PageIndex{6}\):
ΔABC and ΔXYZ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.
Solution
| Step 1. Read the problem. Draw the figure and label it with the given information. | The figure is provided. |
| Step 2. Identify what you are looking for. | The length of the sides of similar triangles |
| Step 3. Name. Choose a variable to represent it. | Let a = length of the third side of ΔABC, y = length of the third side ΔXYZ |
| Step 4. Translate. | The triangles are similar, so the corresponding sides are in the same ratio. So$$\dfrac{AB}{XY} = \dfrac{BC}{YZ} = \dfrac{AC}{XZ}$$Since the side AB = 4 corresponds to the side XY = 3 , we will use the ratio \(\dfrac{AB}{XY} = \dfrac{4}{3}\) to find the other sides. Be careful to match up corresponding sides correctly.
|
| Step 5. Solve the equation. | $$\begin{split} 3a &= 4(4.5) \qquad \; 4y = 3(3.2) \\ 3a &= 18 \qquad \qquad 4y = 9.6 \\ a &= 6 \qquad \qquad \quad y = 2.4 \end{split}$$ |
| Step 6. Check. | $$\begin{split} \dfrac{4}{3} &\stackrel{?}{=} \dfrac{\textcolor{red}{6}}{4.5} \qquad \qquad \qquad \dfrac{4}{3} \stackrel{?}{=} \dfrac{3.2}{\textcolor{red}{2.4}} \\ 4(4.5) &\stackrel{?}{=} 6(3) \qquad \qquad \; 4(2.4) \stackrel{?}{=} 3.2(3) \\ 18 &= 18\; \checkmark \qquad \qquad \quad \; 9.6 = 9.6\; \checkmark \end{split}$$ |
| Step 7. Answer the question. | The third side of ΔABC is 6 and the third side of ΔXYZ is 2.4. |
Exercise \(\PageIndex{11}\):
ΔABC is similar to ΔXYZ. Find a.
- Answer
-
a = 8
Exercise \(\PageIndex{12}\):
ΔABC is similar to ΔXYZ. Find y.
- Answer
-
y = 22.5
Contributors and Attributions
Lynn Marecek(Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content islicensed underCreative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."
FAQs
What is an example of a2 b2 c2? ›
Examples on a2 + b2 + c2 Formula
Answer: a2 + b2 + c2 = 104. Example 2: Find the value of a2 + b2 + c2 if a + b + c = -3, 1/a + 1/b + 1/c = -2 and abc = 3. Answer: a2 + b2 + c2 = 21.
The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.
How do you solve for C in a2 b2 c2? ›To solve for the hypotenuse, we simply take the square root of both sides of the equation a² + b² = c² and solve for c . When doing so, we get c = √(a² + b²) . This is just a reformulation of the Pythagorean theorem and often is not associated with the name hypotenuse formula.
How do you identify a2 +b2? ›The a² + b² Formula is a² + b² = (a +b)² - 2ab and also written as, a² + b² = (a -b)² + 2ab.
Is a2 b2 c2 only for right triangles? ›Pythagorean theorem: If a triangle is a right triangle (has a right angle), then a2+b2=c2. Converse: If a2+b2=c2 in a triangle with c is the longest side, then a triangle is a right triangle. If a triangle is not a right triangle, there are 2 other options for types of triangles.
What is Pythagorean theorem notes 7th grade? ›Pythagorean Theorem DEFINE
In a right triangle, the square of the longest side is equal to the sum of the squares of the shorter two sides. This is represented by the formula a2 + b2= c2.
In a right triangle, a 2 + b 2 = c 2, where a and b are the lengths of the legs and c is the length of the hypotenuse. This is called the Pythagorean theorem.
What is Pythagorean theorem answers? ›For any right triangle, the sum of the squares of the legs of the triangle is equal to the square of the hypotenuse, that is, a2 + b2 = c2. This relationship is known as the PYTHAGOREAN THEOREM. In words, the theorem states that: (leg)2 + (leg)2 = (hypotenuse)2.
What is the formula of A +B +C 2? ›5. (a + b + c)2 = a2 + b2 + c2 +2ab+2bc +2ca.
What is A2 in algebra? ›A2 Maths is the name we give to the second year of A-Level Mathematics. For the full A-Level in Maths, students must study for two full years and take exams in the summer at the end of Year 2.
What is the A square B square formula? ›
a2 + b2 formula is known as the sum of squares formula; it is read as a square plus b square. Its expansion is expressed as a2 + b2= (a + b)2 -2ab.
What is a2 and b2 in Pythagorean theorem? ›The Pythagorean Theorem describes the relationship among the three sides of a right triangle. In any right triangle, the sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse: a2 + b2 = c2.
Is A2 and B2 the same? ›The biggest difference is with the peak brightness, as the B2 gets much brighter, delivering a better HDR experience. The B2 also has many gaming features that the A2 doesn't have, like HDMI 2.1 bandwidth and VRR support.
What is the answer to A2 B2? ›Hence, a2 + b2 is equal to (a + b)² – 2ab.
Is A2 B2 a difference of two square? ›We can represent the general difference of two squares as A2 – B2 Since a number squared is multiplying it by itself, Odd x Even is never possible. Therefore, using rule 4, we can see that if one of A and B is odd, and the other even, the difference of two squares will be Odd.
What is the Pythagorean theorem for dummies? ›Key points. Pythagoras' theorem states that for any right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. It can be thought of as ² + ² = ² where and are the shorter sides of the triangle, and is the hypotenuse (longest side).
What is Pythagorean theorem with example? ›Pythagoras theorem can be used to find the unknown side of a right-angled triangle. For example, if two legs of a right-angled triangle are given as 4 units and 6 units, then the hypotenuse (the third side) can be calculated using the formula, c2 = a2 + b2; where 'c' is the hypotenuse and 'a' and 'b' are the two legs.
What is the Pythagorean theorem for Grade 8? ›Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.
What is the Pythagorean Theorem 4th grade? ›The Pythagorean Theorem can be stated: If a triangle is a right triangle, then a2 + b2 = c2, where a and b represent the lengths of the legs of the right triangle, and c represents the length of the hypotenuse.
What is Pythagorean Theorem 10th grade? ›In a right triangle, a 2 + b 2 = c 2, where a and b are the lengths of the legs and c is the length of the hypotenuse. This is called the Pythagorean theorem.
What grade do kids learn Pythagorean Theorem? ›
Geometry.
In 8th grade, they learn the Pythagorean Theorem and apply it to real-world and mathematical problems.
The Theorem The Pythagorean Theorem states that a2 + b2 = c2, where a and b are the legs and c is the hypotenuse.
What is the formula of degree 2? ›In Maths, the quadratic equation is called a second-degree equation. A quadratic equation is defined as the polynomial equation of the second degree with the standard form ax2 + bx+ c =0, where a≠0, The solutions obtained from the equation are called roots of the quadratic equation.
What is the formula y ax 2 bx c called? ›The graph of a quadratic equation in two variables (y = ax2 + bx + c ) is called a parabola.
What is B in a formula? ›y = mx + b is the slope-intercept form of the equation of a straight line. In the equation y = mx + b, m is the slope of the line and b is the intercept. x and y represent the distance of the line from the x-axis and y-axis, respectively. The value of b is equal to y when x = 0, and m shows how steep the line is.
How long is maths paper 3? ›GCSE Maths Test Paper 3 (calculator)
Making up the remaining 33⅓% of the overall GCSE maths assessment, this paper is once again marked out of 80 and lasts 90 minutes. Students are allowed to use a calculator with this paper. For more information on the GCSE maths test format, please visit the AQA website.
Algebra 2B is a course that completes the 2nd half of the Algebra 2 book. Topics covered will be polynomial functions, function notation, inverses of functions, transformations, exponential growth, and decay, trigonometry, probability, statistics, radical functions, rational functions, sequences, and series.
Is algebra 2 a class? ›In high school, Algebra II helps students gain an understanding of statistics and probability, exponents and logarithms, and mathematical modeling. In general, the Algebra II course covers components in four critical areas: functions, collecting and analyzing data, periodic phenomena, and polynomials.
What is the formula of alpha square minus beta square? ›The a2 - b2 formula is given as: a2 - b2 = (a - b) (a + b).
What is a minus b whole square formula? ›The area of a square whose side is (a – b) is equal to (a – b) square. The area of the horizontal strip (rectangle) equals length multiplied by width, which equals (a – b) times width, which equals b(a – b).
How do you find the square root of a number? ›
What is the Formula for Calculating the Square Root of a Number? The square root of any number can be expressed using the formula: √y = y½. In other words, if a number has 1/2 as its exponent, it means we need to find the square root of the number.
What is a A2 b2 c2? ›a2 + b2 = c2. This is known as the Pythagorean equation, named after the ancient Greek thinker Pythagoras. This relationship is useful because if two sides of a right triangle are known, the Pythagorean theorem can be used to determine the length of the third side.
What is A2 in grammar test? ›An A2 Key qualification is proof of your ability to use English to communicate in simple situations. The exam tests all four English language skills – reading, writing, listening and speaking. It should give you the confidence to go on and study for higher-level exams such as B1 Preliminary and B2 First.
What are the angle measures in a 30 60 90 right triangles? ›A 30-60-90 triangle is a right triangle with angle measures of 30º, 60º, and 90º (the right angle). Because the angles are always in that ratio, the sides are also always in the same ratio to each other.
What is C in a triangle? ›The legs of a right triangle are commonly labeled "a" and "b," while the hypotenuse is labeled "c." Right angle.
What is the 10 example of Pythagorean triplet? ›| (3, 4, 5) | (5, 12, 13) | (8, 15, 17) |
|---|---|---|
| (20, 21, 29) | (12, 35, 37) | (9, 40, 41) |
| (11, 60, 61) | (16, 63, 65) | (33, 56, 65) |
| (13, 84, 85) |
, are (3, 4, 5), (6, 8,10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (16, 30, 34), (21, 28, 35), ... (OEIS A046083, A046084, and A009000).
Is Pythagorean theorem algebra? ›Pythagorean theorem is super important for math. You will probably learn about it for the first time in Algebra, but you will literally use it in Algebra, Geometry, Trigonometry, Precalculus, Calculus, and beyond!
Is The Pythagorean theorem reversible? ›The Theorem is reversible which means that a triangle whose sides satisfy the equality is necessarily right angled. Euclid was the first to mention and prove this fact. 2. The Pythagorean Theorem generalizes to spaces of higher dimensions.