Using the Properties of Triangles to Solve Problems
Learning Outcomes
- Given the measures of two angles of a triangle, find the third
- Use properties of similar triangles to find unknown side lengths
What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle below 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.
ΔABC[/latex]hasvertices[latex]A,B, and C[/latex]andsides[latex]a,b, and c.
The three angles of a triangle are related in a special way. The sum of their measures is 180∘.
m∠A+m∠B+m∠C=180∘
Sum of the Measures of the Angles of a Triangle
For any
ΔABC, the sum of the measures of the angles is
180∘.
m∠A+m∠B+m∠C=180∘
example
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. |
![.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24223631/CNX_BMath_Figure_09_03_050_img-01.png) |
Step 2. Identify what you are looking for. |
The measure of the third angle in the triangle. |
Step 3. Name. Choose a variable to represent it. |
Let x=the measure of the angle. |
Step 4. Translate.
Write the appropriate formula and substitute. |
m∠A+m∠B+m∠C=180 |
Step 5. Solve the equation. |
55°+82°+x=180°
137°+x=180°
x=43° |
Step 6. Check:
55°+82°+43°=?180°
180°=180°✓ |
|
Step 7. Answer the question. |
The measure of the third angle is 43° |
try it
[ohm_question]146498[/ohm_question]
In the following video we show an example of how to find the measure of an unknown angle in a triangle. In this example, we have two triangles who share a common side, and find two unknown interior angles.
https://youtu.be/3kRLkbU6-cI
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 the triangle below.
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
One angle of a right triangle measures
28∘. What is the measure of the third angle?
Answer:
Solution
Step 1. Read the problem. Draw the figure and label it with the given information. |
![.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24223647/CNX_BMath_Figure_09_03_051_img-01.png) |
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 and substitute. |
m∠A+m∠B+m∠C=180° |
Step 5. Solve the equation. |
x+90°+28°=180°
x+118°=180°
x=62° |
Step 6. Check:
180°=?90°+28°+62°
180°=180°✓ |
|
Step 7. Answer the question. |
The measure of the third angle is 62° |
try it
[ohm_question]146499[/ohm_question]
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
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.
Answer:
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).
![.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24223706/CNX_BMath_Figure_09_03_052_img-04.png) |
Step 4. Translate.
Write the appropriate formula and substitute into the formula. |
m∠A+m∠B+m∠C=180
a+(a+20)+90=180 |
Step 5. Solve the equation. |
2a+110=180
2a=70
a=35 first angle
a+20 second angle
35+20
55
90 third angle. |
Step 6. Check:
35°+55°+90°=?180°
180°=180°✓ |
|
Step 7. Answer the question. |
The three angles measure 35°,55°,90° |
try it
[ohm_question]146500[/ohm_question]
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 below are similar. Each side of ΔABC is four times the length of the corresponding side of ΔXYZ and their corresponding angles have equal measures.
ΔABC and ΔXYZ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.
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 BCthe 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
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.
Answer:
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[/latex]=lengthofthethirdsideof[latex]ΔABC
y = length of the third side of ΔXYZ |
Step 4. Translate. |
The triangles are similar, so the corresponding sides are in the same ratio. So
XYAB=YZBC=XZAC
Since the side AB=4 corresponds to the side XY=3 , we will use the ratio XYAB=34 to find the other sides.
Be careful to match up corresponding sides correctly.
![.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24223728/CNX_BMath_Figure_09_03_057_img-01.png) |
Step 5. Solve the equation. |
![.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24223730/CNX_BMath_Figure_09_03_057_img-02.png) |
Step 6. Check:
![.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24223731/CNX_BMath_Figure_09_03_057_img-03.png) |
|
Step 7. Answer the question. |
The third side of ΔABC is 6 and the third side of ΔXYZ is 2.4. |
try it
[ohm_question]146912[/ohm_question]
In the video below we show an example of how to find the missing sides of two triangles that are similar. Note that the measures of the sides in this example are whole numbers, and we use a cross product to solve the resulting proportions.
https://youtu.be/FbtCUXgVA3A
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Licenses & Attributions
CC licensed content, Original
- Question ID 146912, 146498, 146499, 146500. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex 2B: Find the Measure of an Interior Angle of a Triangle. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757.