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7.04 Congruence in right triangles

Lesson

Concept summary

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 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.

Worked examples

Example 1

Identify the additional information needed to show these triangles are congruent by hypotenuse-leg congruence.

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.

Approach

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.

Solution

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

Reflection

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

Example 2

Consider the following:

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

Show that \triangle{ILK}\cong \triangle{JLK} and justify your reasoning.

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.

Approach

When showing triangles are 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 show that two triangles are congruent, we need to decide which triangle congruence theorem best fits the given information. Then, when writing our explanation we always start with the given information, build any conclusions based on the given information, and end with the conclusion we are trying to show.

Solution

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 hypotenuse-leg theorem.

Reflection

When writing explanations justifying triangle congruence, it is important that each statement you make to show congruence is justified.

Outcomes

M2.G.CO.B.6

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.

M2.G.CO.B.7

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

M2.G.SRT.B.3

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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