Topics
10. Congruence and Triangles
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United States of America
Grades 9-12

10.05 Right triangle congruence

Lesson
Practice

Introduction

In this final lesson on congruence criteria for triangles, we explore a theorem specific to right triangles, convincing ourselves that the criteria presented will only work for right triangles and proving congruence using rigid motions before formalizing the theorem.

Right triangle congruence

Exploration

Tawana says that the congruence theorem: Side-Side-Angle (SSA) is a valid congruence theorem and works for oblique triangles.

  1. Explain why SSA is not valid by providing evidence countering Tawana's claim.

  2. Tawana states that SSA is valid, however, for right triangles. Explain why this statement made by Tawana is valid.

In general, side-side-angle is not a valid congruence theorem. When the congruent angle pair are right angles, however, we introduce a new triangle congruence theorem specific to right triangles:

Hypotenuse-Leg (HL) congruency theorem

If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent

Remember, the hypotenuse is the side opposite the right angle.

Two right triangles, triangle A B C with right angle B and triangle D E F with right angle E are drawn such that segment A B and segment D E are congruent as well as segment A C and segment D F.

Examples

Example 1

Prove the HL congruency theorem using rigid transformations with the given diagram.

Given:

  • \overline{RT}\cong \overline{XZ}
  • \overline{ST}\cong \overline{YZ}
  • m \angle RST = 90 \degree \text{ and } m \angle XYZ= 90 \degree

Prove: \triangle RST \cong \triangle XYZ

Worked Solution
Apply the idea

1. There is a rigid motion that will map \angle XYZ to \angle RST because \angle RST \cong \angle XYZ. First, we can rotate \triangle XYZ clockwise about Y until \overline{YZ} \parallel \overline{ST}.

2. Since the transformation is a rigid motion, we have that: \angle XYZ \cong \angle X'Y'Z' \text{, } \overline{XZ} \cong \overline {X'Z'} \text{, and } \overline{YZ} \cong \overline{Y'Z'}

3. Using the given information and the transitive property of congruence, we have that: \angle RST \cong \angle X'Y'Z' \text{, } \overline{RT} \cong \overline {X'Z'} \text{, and } \overline{ST} \cong \overline{Y'Z'}

4. Finally, draw a vector from X to R, and translate \triangle XYZ along the vector. A translation preserves angle measures and side lengths.

We have now shown that:

  • X maps to R using rigid motions
  • Y maps to S using rigid motions
  • Z maps to T using rigid motions

So we have that \triangle XYZ can be mapped onto \triangle RST, so \triangle RST \cong \triangle XYZ.

Reflect and check

We also used rigid transformations and the transitive property of congruence to show that triangles were congruent in lesson  5.03 SSS and SAS congruence criteria  and lesson  5.04 ASA and AAS congruence criteria  .

Example 2

Given:

  • \overline{IL}\cong \overline{JL}
  • \overline{LK} \perp \overline{IJ}

Prove: \triangle{ILK}\cong \triangle{JLK}

Isosceles Triangle I L J with congruent segments I L and J L is drawn. Segment L K is drawn intersecting segment I J at point K.
Worked Solution
Create a strategy

When proving triangles congruent, start by labeling and identifying the given information as well as anything that we can conclude directly from the diagram (such as vertical angles or reflexively congruent parts). Since we are trying to prove two triangles congruent, we need to decide which triangle congruence theorem best fits the given information. Then, when writing the proof we always start with the given information, build any conclusions based on the given information, and end with the conclusion we are trying to prove.

Apply the idea

We are given that \overline{LK} \perp \overline{IJ} which means that \angle{IKL} and \angle{JKL} are both right angles by the definition of perpendicular lines. That means \triangle{ILK} and \triangle{JLK} are both right triangles by the definition of right triangles. We are also given that \overline{IL}\cong \overline{JL} which means we have two congruent hypotenuses, and we know that \overline{LK}\cong \overline{LK} by the reflexive property so we have a pair of congruent legs as well. This is enough information to conclude that \triangle{ILK}\cong \triangle{JLK} by the HL theorem.

Reflect and check

Triangle proofs can be written in any style - just make sure that each congruent part needed to justify the triangle congruence theorem has its own statement and reason included.

Example 3

Consider the triangles shown below:

Two right triangles, triangle M G H with right angle G and triangle X N S with right angle N are drawn such that segment M G is congruent to segment X N.
a

Identify the additional information needed to prove these triangles congruent by HL congruence.

Worked Solution
Create a strategy

From the diagram we know that \angle{G} and \angle{N} are both right angles making both triangles right triangles. We also know that \overline{MG} \cong \overline{NX} which is a pair of corresponding legs. We are missing congruent hypotenuses to complete the theorem.

Apply the idea

\overline{MH}\cong \overline{XS}

Reflect and check

Having a right angle alone is not enough to justify HL congruency. Make sure that the hypotenuses are congruent and one pair of legs are also congruent.

b

What other information could be given about the triangles to be able to prove congruence?

Worked Solution
Create a strategy

By taking the original diagram into consideration, we can seek to use congruence between unknown corresponding angles to show that the triangles are congruent.

Apply the idea

If we are given that \angle M \cong \angle X, we can prove the two triangles congruent by ASA congruence.

Example 4

Find the value for each of the variables that makes \triangle{ABC} \cong \triangle{CDE}.

Worked Solution
Create a strategy

Since m \angle ABC and m \angle CDE = 90 \degree, we know that \triangle ABC and \triangle CDE are both right triangles.

We know that if the hypotenuse and leg of one triangle are congruent to the hypotenuse and leg of another triangle, then the triangles are congruent. We need to choose values for x and y that will make the hypotenuses and the legs congruent.

We can use that \overline{AC} \cong \overline{CE} and \overline{AB} \cong \overline{CD} to solve for the variables. By the definition of congruence, since \overline{AC} \cong \overline{CE} and \overline{AB} \cong \overline{CD}, the two pairs of segments will be equal in length.

Apply the idea
\displaystyle \dfrac{3}{2}x-4\displaystyle =\displaystyle \dfrac{1}{2}x+7AC=CE
\displaystyle x-4\displaystyle =\displaystyle 7Subtract \dfrac{1}{2}x from both sides
\displaystyle x\displaystyle =\displaystyle 11Add 4 to both sides
\displaystyle \dfrac{3y+2}{4}\displaystyle =\displaystyle 8AB=CD
\displaystyle 3y+2\displaystyle =\displaystyle 32Multiply both sides by 4
\displaystyle 3y\displaystyle =\displaystyle 30Subtract 2 from both sides
\displaystyle y\displaystyle =\displaystyle 10Divide both sides by 3
Idea summary

To show that two right triangles are congruent, it is sufficient to demonstrate Hypotenuse-leg, (or HL): If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent

Outcomes

G.CO.B.7

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G.CO.B.8

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G.CO.C.9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

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