Section 4.3Congruent Triangles

If two geometric figures have exactly the same shape and size, they are congruent.

In two congruent polygons, all of the parts of one polygon are congruent to the corresponding parts or matching parts of the other polygon. These corresponding parts include corresponding angles and corresponding sides.

Concept 1

Other congruence statements for the triangles above exist. Valid congruence statements for congruent polygons list corresponding vertices in the same order.

Example 1:

a) Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.

Angles: A R, B T, C P, D S, E Q

Sides: , , , ,AB RT BC TP CD PS DE SQ EA QR

All corresponding parts of the two polygons are congruent. Therefore, ABCDE RTPSQ.

b) The support beams on the fence form congruent triangles. In the figure ΔABC ΔDEF, which of the following congruence statements correctly identifies corresponding angles or sides?

a)

b)

c)

d)

The phrase “if and only if” in the congruent polygon definition means that both the conditional and converse are true. So, if two polygons are congruent, then their corresponding parts are congruent.  

For triangles we say Corresponding Parts of Congruent Triangles are Congruent or CPCTC.

Example 2:

a) In the diagram, ΔITP ΔNGO. Find the values of x and y.

O PCPCTCmO = mP Definition of congruence6y – 14= 40 Substitution

6y = 54 Add 14 to each sidey = 9 Divide each side by 6

NG = IT Definition of congruencex – 2y = 7.5 Substitutionx – 2(9) = 7.5 y = 9x – 18 = 7.5 Simplifyx = 25.5 Add 18 to each side

CPCTCNG IT

x = 25.5, y = 9

b) In the diagram, ΔFHJ ΔHFG. Find the values of x and y.

GFH HFG so:6x + 8 = 35, solve for x.x = 4.5

so:

2 3 2.5, solve for .

2.75

FJ HG

FJ HG

y y

y

Concept 2

Example 3: ARCHITECTURE A drawing of a tower’s roof is composed of congruent triangles all converging at a point at the top. If IJK IKJ and mIJK = 72, find mJIH.

mIJK + mIKJ + mJIK = 180° Triangle Angle-Sum TheoremΔJIK ΔJIH Congruent Triangles

mIJK + mIJK + mJIK = 180° Substitution72° + 72° + mJIK = 180° Substitution

144° + mJIK = 180° SimplifymJIK = 36° Subtract 144

from each sidemJIH = 36° Third Angles

Theorem

Example 4: Write a two-column proof.

Prove: ΔLMN ΔPON

Given: ÐL @ ÐPLM POLN PNMN NO

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

ÐL @ ÐP Given

LM PO

LN PN

MN NO

Given

Given

Given

LNM PNO

M OΔLMN ΔPON

Vertical Angles Theorem

Third Angles Theorem

CPCTC

Concept 3