6.11: Identify and Apply Theorems to Triangle Congruence
Have you ever made a kite? Take a look at this dilemma.
Justin wants to make a kite. To help, he created the following pattern.
Justin needs to be sure that each triangle is congruent. He knows that there is a way to do this, but he can't remember how to figure this out.
Pay attention and this Concept will teach you what you need to know to figure out whether Justin's triangles are congruent.
Guidance
You can identify the congruent sides and angles of polygons and use that to determine congruence.
If the sides and the angles of any two polygons are congruent, then we know that the two polygons are also congruent.
We can also work with triangles, because after all a triangle is also a polygon. Triangles are unique because there are a few rules that we can use to help us to identify whether or not two triangles are congruent. You will notice that if you learn these rules, that you won’t have to compare every angle and every side to determine whether or not two triangles are congruent.
The first rule represents the sidesideside, or SSS, relationship. It says that if each of the three sides of one triangle is congruent to a side in a second triangle, then the two triangles are congruent. We do not need to check the angles. In triangles, angles are always in a fixed relationship with the side opposite them. The wider the angle, the longer the side opposite it must be. If we know the sides are congruent, then we know the angles must be also.
The second rule says that if one angle and the sides adjacent to it in one triangle are congruent to an angle and its adjacent sides in the second triangle, the triangles will be congruent. We call this the sideangleside (SAS) rule. In other words, this time we only need to make sure one angle and two sides are congruent to know that all parts of the triangles are congruent. However, remember that the angle you use must be located between the two sides.
The third rule tells us that if two angles and the side between them in one triangle are congruent to two angles and the side between them in the second triangle, the two triangles are congruent. This is the anglesideangle (ASA) rule. We can determine congruence just by knowing two angles and one side.
Are the triangles below congruent? Explain your reasoning.
Now let’s start by looking at the triangles and looking for the given information. We know that we need three side measurements for SSS, or a side measurement, an angle measurement and a side measurement for SAS, or an angle side angle measurement for ASA.
We know that two pairs of sides match: one pair is 7.5 centimeters and the other is 4 centimeters. This is not enough information to know for sure that the triangles themselves are congruent, because the sides may be in different places. Which rule can we use to check for congruence? We cannot use SSS because we only know the lengths of two sides.
We can use SAS, however. Remember, to use SAS, the angle must be between the two sides. In the first triangle, the angle between the two sides is a right angle, so we know that it is \begin{align*}90^\circ\end{align*}
Are these two triangles congruent? Explain your reasoning.
Here we have two triangles with given side lengths. We can see that these two triangles are congruent because their side lengths are congruent. The side lengths of both triangles are labeled and we can prove they are congruent by applying the SSS rule.
You can see how helpful these rules are in thinking about triangles and their congruence.
Name the theorem that you could use to prove triangle congruence based on each description.
Example A
You have been given two side lengths and an angle measure.
Solution: SAS
Example B
You have been given two angle measures and one side length.
Solution: AAS
Example C
You have been given three side lengths and no angle measures.
Solution: SSS
Now let's go back to the dilemma from the beginning of the Concept.
Here is the pattern that Justin drew for his kite.
Now let's think about which theorem we can use to prove that the two triangles are congruent. These triangles share a side, so we know that that side is congruent to both triangles. The tic marks show that the other two sides are congruent as well.
We can say that the SSS theorem proves that these triangles are congruent.
Vocabulary
 Congruent
 exactly the same, having the same size, shape and measurement.
 Corresponding parts
 When two figures are congruent, there are matching parts for each of the two figures.
 SSS
 determining triangle congruence by comparing the three side lengths of two triangles.
 SAS
 determining triangle congruence by comparing the side, angle, side of two triangles.
 ASA
 determining triangle congruence by comparing the angle, side, angle of two triangles.
Guided Practice
Here is one for you to try on your own.
Use this illustration to answer the following question.
Which theorem could you use to prove that these triangles are congruent? Explain your thinking.
Solution
First, notice that we haven't been given any angle measures or specific side lengths. However, the tic marks show that the side lengths in triangle ABC have a corresponding side length of the same measure in angle LKM.
Therefore, we can use the SSS theorem to prove that these triangles are congruent.
Video Review
Congruent and Similar Triangles
Practice
Directions: Use the given information to figure out each congruence statement.
\begin{align*}\Delta ABC \cong \Delta DEF\end{align*}

\begin{align*}\angle A \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
∠A≅−−−−−  \begin{align*}\angle B \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}\angle C \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}\overline{AB} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}\overline{BC} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}\overline{AC} \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 If line segment \begin{align*}AC\end{align*} has a length of 8, which other segment also has a length of 8?
 If angle \begin{align*}A\end{align*} has a measure of \begin{align*}55^\circ\end{align*}, which other angle has a measure of \begin{align*}55^\circ\end{align*}?
 If angle \begin{align*}B\end{align*} has a measure of \begin{align*}45^\circ\end{align*}, which other angle has a measure congruent to that?
 If these two triangles are congruent, are the side lengths and angle measures the same?
 Would these two triangles look identical?
 Which two letters would represent the vertex of each triangle?
Directions:Name the theorem that would best prove triangle congruence based on each description.
13. Three side lengths, 6 inches, 5 inches and 4 inches.
14. One angle measure and two side lengths.
15. Two angle measures and one side length
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AAS (AngleAngleSide)
If two angles and a nonincluded side in one triangle are congruent to two angles and the corresponding nonincluded side in another triangle, then the triangles are congruent.Angle Side Angle Triangle
The term 'anglesideangle triangle' refers to a triangle with known measures of two angles and the length of the side between them.ASA
ASA, anglesideangle, refers to two known angles in a triangle with one known side between the known angles.Congruent
Congruent figures are identical in size, shape and measure.Corresponding Angles
Corresponding angles are two angles that are in the same position with respect to the transversal, but on different lines.Reflexive Property of Congruence
orSAS
SAS means side, angle, side, and refers to the fact that two sides and the included angle of a triangle are known.Side Side Side Triangle
A side side side triangle is a triangle where the lengths of all three sides are known quantities.SSS
SSS means side, side, side and refers to the fact that all three sides of a triangle are known in a problem.Image Attributions
Here you'll identify and apply theorems to test triangle congruence.