Solve each equation.

1.x + 6 = 25 2. x + 7 + 13 = 33

3.5x = 540 4. x + 10 = 2x

5.For the triangle at the right, use the Triangle

Angle-Sum Theorem to find the value of y.

(For help, go to Algebra Review, page 30.)

GEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

Congruent FiguresCongruent Figures

4-1

Check Skills You’ll Need

x = 19 x = 13

x = 108 x = 10

y = 50

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

4-1

Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides.

Two polygons are congruent polygons if and only if their corresponding sides and corresponding angles are congruent.

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

4-1

In a congruence statement, the order of the vertices indicates the corresponding parts.

When you write a statement such as ABC DEF, you are also stating which parts are congruent.

Helpful Hint

List the corresponding sides in the same order.

List the corresponding vertices in the same order.

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

ABC QTJ. List the congruent corresponding

parts.

Angles: A Q B T C J

Sides: AB QT BC TJ AC QJ

4-1

Quick Check

Naming Congruent Parts

XYZ KLM, mY = 67, and mM = 48. Find mX.

Use the Triangle Angle-Sum Theorem and the definition of congruent polygons to find mX.

mX = 65 Subtract 115 from each side.

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

mX + mY + mZ = 180 Triangle Angle-Sum Theorem

mZ = mM Corresponding angles of congruenttriangles that are congruent

mZ = 48 Substitute 48 for mM.

mX + 67 + 48 = 180 Substitute.

mX + 115 = 180 Simplify.

4-1

Quick Check

Using Congruent Polygons

Can you conclude that ABC CDE in the figure below?

List corresponding vertices in the same order.

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

If ABC CDE, then BAC DCE.

The diagram above shows BAC DEC, not DCE.

4-1

Corresponding angles are not necessarily congruent, therefore you

cannot conclude that ABC CDE.Quick Check

Finding Congruent Triangles

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

4-1

Theorem 4-1

Show how you can conclude that CNG DNG. List statements and reasons.

Congruent triangles have three congruent corresponding sides and three congruent corresponding angles.Examine the diagram, and list the congruent corresponding parts for CNG and DNG.

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

a. CG DG Givenb. CN DN Givenc. GN GN Reflexive Property of Congruenced. C D Givene. CNG DNG Right angles are congruent.

f. CGN DGN Third Angles Theorem

g. CNG DNG Definition of triangles

4-1

Quick Check

Proving Triangles Congruent

In Exercises 1 and 2, quadrilateral WASH quadrilateral NOTE.

1. List the congruent corresponding parts.

2. mO = mT = 90 and mH = 36. Find mN.

3. Write a statement of triangle congruence.

4. Write a statement of triangle congruence.

5. Explain your reasoning in Exercise 4 above.

144

Congruent FiguresCongruent FiguresGEOMETRY LESSON 4-1GEOMETRY LESSON 4-1

Sample: Two pairs of corresponding sides and two pairs of corresponding angles are given. C A because all right angles are congruent. BD BD by the Reflexive Property of . ABD CDB by the definition of congruent triangles.

Sample: DFH ZPR

Sample: ABD CDB

4-1

WA NO, AS OT, SH TE, WH NE;W N, A O, S T, H E