This document provides information about using the Hypotenuse-Leg Theorem to prove triangles congruent. It gives examples of when the theorem can and cannot be used. It also asks what additional information is needed to prove triangles congruent using the theorem. The document then discusses how to identify and separate overlapping triangles when proving congruence. It provides examples of writing paragraph proofs to show triangles are congruent using angle-angle-side, side-side-side, and angle-side-angle criteria.

1. UNIT 4.7 CONGRUENCE INUNIT 4.7 CONGRUENCE IN
RIGHT TRIANGLESRIGHT TRIANGLES

2. For Exercises 1 and 2, tell whether the HL Theorem can be used to prove
the triangles congruent. If so, explain. If not, write not possible.
1. 2.
For Exercises 3 and 4, what additional information do you
need to prove the triangles congruent by the HL Theorem?
3. LMX LOX 4. AMD CNB
Not
possible
Yes; use the congruent
hypotenuses and leg BC
to prove ABC DCB
LM LO AM CN
or
MD NB

3. 1.How many triangles will the next two figures in this pattern have?
2.Can you conclude that the triangles are congruent? Explain.
a. AZK and DRS
b. SDR and JTN
c. ZKA and NJT
For every new right triangle, segments connect the midpoint of the
hypotenuse with the midpoints of the legs of the right triangle,
creating two new triangles for every previous new triangle. The first
figure has 1 triangle. The second has 1 + 2, or 3 triangles. The third
has 3 + 4, or 7 triangles. The fourth will have 7 + 8, or 15 triangles.
The fifth will have 15 + 16, or 31 triangles.
a. Two pairs of sides are
congruent. The included
angles are congruent. Thus,
the two triangles are
congruent by SAS.
b. Two pairs of angles are
congruent. One pair of
sides is also congruent,
and, since it is opposite a
pair of corresponding
congruent angles, the
triangles are congruent by
AAS.
c. Since ∆AZK ≅ ∆DRS
and ∆SDR ≅ ∆JTN, by
the Transitive Property
of ≅, ∆ZKA ≅ ∆NJT.

4. Some triangle relationships are difficult to see
because the triangles overlap.
Overlapping triangles may have a common side or
angle. You can simplify your work with overlapping
triangles by separating and redrawing the triangles.
Overlapping triangles share part or all of one or
more sides.

5. Name the parts of their sides that ∆DFG and ∆EHG share.
These parts are HG and FG, respectively.
Parts of sides DG and EG are shared by DFG and EHG.
Identify the overlapping triangles.
Identifying Common Parts

6. Write a Plan for Proof that does not use overlapping triangles.
Given: ∠ZXW ≅ ∠YWX, ∠ZWX ≅ ∠YXW
Prove: ZW ≅ YX
You can prove these triangles congruent using ASA as follows:
Label point M where ZX intersects WY, as shown in the
diagram. ZW YX by CPCTC if ZWM YXM.
Look at MWX. MW MX by the Converse of the Isosceles Triangle
Theorem.
Look again at ZWM and YXM. ∠ ZMW ∠YMX because vertical
angles are congruent, MW MX, and by subtraction ∠ ZWM ∠YXM, so
ZWM YXM by ASA.
Planning a Proof

7. Write a paragraph proof.
Given: XW YZ, ∠XWZ and ∠YZW are right angles.
Prove: XPW YPZ
Plan: XPW YPZ by AAS if ∠WXZ ∠ZYW. These
angles are congruent by CPCTC if XWZ YZW.
These triangles are congruent by SAS.
Proof: You are given XW YZ. Because ∠XWZ and ∠YZW are right angles,
∠XWZ ∠YZW. WZ ZW, by the Reflexive Property of Congruence.
Therefore, XWZ YZW by SAS. ∠WXZ ∠ZYW by CPCTC, and
∠XPW ∠YPZ because vertical angles are congruent.
Therefore, XPW YPZ by AAS.
Using Two Pairs of Triangles

8. Given: CA CE, BA DE
Write a two-column proof to show that ∠CBE ∠CDA.
3. CA = CE, BA = DE 3. Definition of congruent segments.
4. CA – BA = CE – DE 4. Subtraction Property of Equality
5. CA – BA = CB, 5. Segment Addition Postulate
CE – DE = CD
6. CB = CD 6. Substitution
Plan: ∠CBE ∠CDA by CPCTC if CBE CDA. This
congruence holds by SAS if CB CD.
Proof: Statements Reasons
1. ∠BCE ∠DCA 1. Reflexive Property of Congruence
2. CA CE, BA DE 2. Given
7. CB CD 7. Definition of congruence
8. CBE CDA 8. SAS
9. ∠CBE ∠CDA 9. CPCTC
Separating Overlapping Triangles

9. 4-7
GEOMETRY LESSON 4-7
1. Identify any common sides and
angles in AXY and BYX.
For Exercises 2 and 3, name a pair of
congruent overlapping triangles. State the
theorem or postulate that proves them congruent.
2. 3.
4. Plan a proof.
Given: AC BD, AD BC
Prove: XD XC
XY
KSR MRS
SAS
GHI IJG
ASA
XD XC by CPCTC if DXA CXB.
This congruence holds by AAS if
BAD ABC. Show BAD ABC
by SSS.

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