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Lesson 4-1 Classifying Triangles

Lesson 4-2 Angles of Triangles

Lesson 4-3 Congruent Triangles

Lesson 4-4 Proving Congruence–SSS, SAS

Lesson 4-5 Proving Congruence–ASA, AAS

Lesson 4-6 Isosceles Triangles

Lesson 4-7 Triangles and Coordinate Proof

Example 1 Classify Triangles by Angles

Example 2 Classify Triangles by Sides

Example 3 Find Missing Values

Example 4 Use the Distance Formula

ARCHITECTURE The triangular truss below is modeled for steel construction. ClassifyJMN, JKO, and OLN as acute, equiangular, obtuse, or right.

Answer:JMN has one angle with measure greater than 90, so it is an obtuse triangle.

JKO has one angle with measure equal to 90, so it is a right triangle.

OLN is an acute triangle with all angles congruent, so it is an equiangular triangle.

ARCHITECTURE The frame of this window design is made up of many triangles. Classify ABC, ACD, and ADE as acute, equiangular, obtuse, or right.

Answer: ABC is acute. ACD is obtuse. ADE is right.

Answer: UTX and UVX are isosceles.

Identify the isosceles triangles in the figure if

Isosceles triangles have at least two sides congruent.

Identify the scalene triangles in the figure if

Answer: VYX, ZTX, VZU, YTU, VWX, ZUX, and YXU are scalene.

Scalene triangles have no congruent sides.

Identify the indicated triangles in the figure.

a. isosceles triangles

b. scalene triangles

Answer: ABC, EBC, DEB, DCE, ADC, ABD

Answer: ADE, ABE

ALGEBRA Find d and the measure of each side of equilateral triangle KLM if and

Since KLM is equilateral, each side has the same length. So

Substitution

Subtract d from each side.

Add 13 to each side.

Divide each side by 3.

Next, substitute to find the length of each side.

Answer: For KLM, and the measure of each side is 7.

Answer:

ALGEBRA Find d and the measure of each side of equilateral triangle if and

COORDINATE GEOMETRY Find the measures of the sides of RST. Classify the triangle by sides.

Answer: ; since all 3 sides have different lengths, RST is scalene.

Use the distance formula to find the lengths of each side.

Find the measures of the sides of ABC. Classify the triangle by sides.

Answer: ; since all 3 sides have different lengths, ABC is scalene.

Example 1 Interior Angles

Example 2 Exterior Angles

Example 3 Right Angles

Find the missing angle measures.

Find first because the measure of two angles of the triangle are known.

Angle Sum Theorem

Simplify.

Subtract 117 from each side.

Answer:

Angle Sum Theorem

Simplify.

Subtract 142 from each side.

Find the missing angle measures.

Answer:

Find the measure of each numbered angle in the figure.

Exterior Angle TheoremSimplify.

SubstitutionSubtract 70 from each side.

If 2 s form a linear pair, they are supplementary.

Exterior Angle Theorem

Subtract 64 from each side.

Substitution

Subtract 78 from each side.

If 2 s form a linear pair, they are supplementary.

Substitution

Simplify.

Subtract 143 from each side.

Angle Sum Theorem

Substitution

Simplify.

Answer:

Find the measure of each numbered angle in the figure.

Answer:

Corollary 4.1

Substitution

Subtract 20 from each side.

Answer:

GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20.

Answer:

The piece of quilt fabric is in the shape of a right triangle. Find if is 32.

Example 1 Corresponding Congruent Parts

Example 2 Transformations in the Coordinate Plane

Answer: Since corresponding parts of congruent triangles are congruent,

ARCHITECTURE A tower roof is composed of congruent triangles all converging toward a point at the top. Name the corresponding congruent angles and sides of HIJ and LIK.

Name the congruent triangles.

ARCHITECTURE A tower roof is composed of congruent triangles all converging toward a point at the top.

Answer: HIJ LIK

Answer:

The support beams on the fence form congruent triangles.

b. Name the congruent triangles.

a. Name the corresponding congruent angles and sides of ABC and DEF.

Answer: ABC DEF

COORDINATE GEOMETRY The vertices of RST are R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST are R(3, 0), S(0, ─5), and T(─1, ─1). Verify that RST RST.

Use the Distance Formula to find the length of each side of the triangles.

Use the Distance Formula to find the length of each side of the triangles.

Use the Distance Formula to find the length of each side of the triangles.

Use a protractor to measure the angles of the triangles. You will find that the measures are the same.

Answer: The lengths of the corresponding sides of two triangles are equal. Therefore, by the definition of congruence,

In conclusion, because ,

COORDINATE GEOMETRY The vertices of RST are R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST are R(3, 0), S(0, ─5), and T(─1, ─1). Name the congruence transformation for RST and RST.

Answer: RST is a turn of RST.

COORDINATE GEOMETRY The vertices of ABC are A(–5, 5), B(0, 3), and C(–4, 1). The vertices of ABC are A(5, –5), B(0, –3), and C(4, –1).

Answer:

Use a protractor to verify that corresponding angles are congruent.

a. Verify that ABC ABC.

Answer: turn

b. Name the congruence transformation for ABC and ABC.

Example 1 Use SSS in Proofs

Example 2 SSS on the Coordinate Plane

Example 3 Use SAS in Proofs

Example 4 Identify Congruent Triangles

ENTOMOLOGY The wings of one type of moth form two triangles. Write a two-column proof to prove thatFEG HIG and G is the midpoint of both

Given: G is the midpoint of both

Prove:

1. Given1.

Proof: ReasonsStatements

3. SSS3. FEG HIG

FEG HIG

2. Midpoint Theorem2.

3. SSS

1. Given2. Reflexive

Proof: ReasonsStatements

1. 2.3. ABC GBC

Write a two-column proof to prove that ABC GBC if

Use the Distance Formula to show that the corresponding sides are congruent.

COORDINATE GEOMETRY Determine whether WDV MLP for D(–5, –1), V(–1, –2), W(–7, –4), L(1, –5), P(2, –1), and M(4, –7). Explain.

Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore, WDV MLP by SSS.

Answer: By definition of congruent segments, all corresponding segments are congruent. Therefore, ABC DEF by SSS.

Determine whether ABC DEF for A(5, 5), B(0, 3), C(–4, 1), D(6, –3), E(1, –1), and F(5, 1). Explain.

Write a flow proof.

Given:

Prove: QRT STR

Answer:

Write a flow proof.

Given: .

Prove: ABC ADC

Proof:

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

Answer: SAS

Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. The triangles are congruent by SAS.

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

Answer: SSS

Each pair of corresponding sides are congruent. Two are given and the third is congruent by Reflexive Property. So the triangles are congruent by SSS.

Answer: SAS

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

a.

Answer: not possible

b.

Example 1 Use ASA in Proofs

Example 2 Use AAS in Proofs

Example 3 Determine if Triangles Are Congruent

Proof: because alternate interior angles are

congruent. By the Midpoint Theorem,

Since vertical angles are congruent,

WRL EDL by ASA.

Write a paragraph proof.

Given: L is the midpoint ofProve: WRL EDL

Proof: because alternate interior

angles are congruent.

because alternate interior angles are congruent.

by Reflexive Property. ABD CDB by ASA.

Write a paragraph proof.

Given:

Prove: ABD CDB

Write a flow proof.

Given:

Prove:

Proof:

Proof:

Given:

Prove:

Write a flow proof.

STANCES When Ms. Gomez puts her hands on her hips, she forms two triangles with her upper body and arms. Suppose her arm lengths AB and DE measure 9 inches, andAC and EF measure 11 inches. Also suppose that you are given that Determine whether ABC EDF. Justify your answer.

Explore We are given measurements of two sides of each triangle. We need to determine whether the two triangles are congruent.

Plan Since Likewise, We are given Check each possibility using the five methods you know.

Answer: ABC EDF by SSS

Solve We are given information about three sides. Since all three pairs of corresponding sides of the triangles are congruent, ABC EDF by SSS.

Examine You can measure each angle in ABC and EDF to verify that

Answer: ABE CBD by SSS

The curtain decorating the window forms 2 triangles at the top. B is the midpoint of AC. inches and

inches. BE and BD each use the same amount of material, 17 inches. Determine whether ABE CBDJustify your answer.

Example 1 Proof of Theorem

Example 2 Find the Measure of a Missing Angle

Example 3 Congruent Segments and Angles

Example 4 Use Properties of Equilateral Triangles

Write a two-column proof.

Given:

Prove:

Proof:

ReasonsStatements

3. Def. of isosceles 3. ABC and BCD are isosceles

1. Given1.

6. 6. Substitution

5. 5. Given

4. 4. Isosceles Theorem

2. Def. of segments2.

Write a two-column proof.

Given: .

Prove:

Proof:

ReasonsStatements

1. Given

3. Isosceles Theorem

2. Def. of isosceles triangles

1.

2. ADB is isosceles.

3.

4.

5.

4. Given

5. Def. of midpoint

6. SAS

7. 7. CPCTC

6. ABC ADC

Multiple-Choice Test Item If and what is the measure of

Read the Test Item CDE is isosceles with base Likewise, CBA is isosceles with

A. 45.5 B. 57.5 C. 68.5 D. 75

Solve the Test Item

Angle Sum Theorem

Substitution

Add.

Subtract 120 from each side.Divide each side by 2.

Step 1 The base angles of CDE are congruent. Let

Def. of vertical anglesSubstitution

Add.

Step 2 are vertical angles so they have equal measures.

Step 3 The base angles of CBA are congruent.

Angle Sum Theorem

Substitution

Subtract 30 from each side.Divide each side by 2.

Answer: D

Answer: A

Multiple-Choice Test Item If and what is the measure of

A. 25 B. 35 C. 50 D. 130

Answer:

Name two congruent angles.

Answer:

Name two congruent segments.

By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So,

a. Name two congruent angles.

Answer:

Answer:

b. Name two congruent segments.

Since the angle was bisected,Each angle of an equilateral triangle measures 60°.

EFG is equilateral, and bisects bisectsFind and

Answer:

Add.

Exterior Angle Theorem

Substitution

is an exterior angle of EGJ.

Subtract 75 from each side.

Linear pairs are supplementary.

Substitution

Answer: 105

EFG is equilateral, and bisects bisectsFind

a. Find x.

b.

Answer: 90

Answer: 30

ABC is an equilateral triangle. bisects

Example 1 Position and Label a Triangle

Example 2 Find the Missing Coordinates

Example 3 Coordinate Proof

Example 4 Classify Triangles

Use the origin as vertex X of the triangle.

Place the base of the triangle along the positive x-axis.

Position and label right triangle XYZ with leg d units long on the coordinate plane.

X (0, 0) Z (d, 0)

Position the triangle in the first quadrant.

Since Z is on the x-axis, its y-coordinate is 0. Its x-coordinate is d because the base is d units long.

Since triangle XYZ is a right triangle the x-coordinate of Y is 0. We cannot determine the y-coordinate so call it b.

Answer:

X (0, 0) Z (d, 0)

Y (0, b)

Answer:

Position and label equilateral triangle ABC with side w units long on the coordinate plane.

Name the missing coordinates of isosceles right triangle QRS.

Answer: Q(0, 0); S(c, c)

Q is on the origin, so its coordinates are (0, 0). The x-coordinate of S is the same as the x-coordinate for R, (c, ?).

The y-coordinate for S is the distance from R to S. Since QRS is an isosceles right triangle,The distance from Q to R is c units. The distance from R to S must be the same. So, the coordinates of S are (c, c).

Answer: C(0, 0); A(0, d)

Name the missing coordinates of isosceles right ABC.

Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base.

Prove:

The first step is to position and label a right triangle on the coordinate plane. Place the base of the isosceles triangle along the x-axis. Draw a line segment from the vertex of the triangle to its base. Label the origin and label the coordinates, using multiples of 2 since the Midpoint Formula takes half the sum of the coordinates.

Given: XYZ is isosceles.

Proof: By the Midpoint Formula, the coordinates of W,

the midpoint of , is

The slope of or undefined. The

slope of is therefore, .

Write a coordinate proof to prove that the segment drawn from the right angle to the midpoint of the hypotenuse of an isosceles right triangle is perpendicular to the hypotenuse.

Proof: The coordinates of the midpoint D are

The slope of is

or 1. The slope of or –1,

therefore .

DRAFTING Write a coordinate proof to prove that the outside of this drafter’s tool is shaped like a right triangle. The length of one side is 10 inches and the length of another side is 5.75 inches.

Proof: The slope of

or undefined. The slope of

or 0, therefore

DEF is a right triangle.

The drafter’s tool is shaped like a

right triangle.

FLAGS Write a coordinate proof to prove this flag is shaped like an isosceles triangle. The length is 16 inches and the height is 10 inches.

C

Proof: Vertex A is at the origin and B is at (0, 10). The x-coordinate of C is 16. The y-coordinate is halfway between 0 and 10 or 5. So, the coordinates of C are (16, 5).

Determine the lengths of CA and CB.

Since each leg is the same length, ABC is isosceles. The flag is shaped like an isosceles triangle.

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