Geometry Concepts

Chapter 5 Triangle and Congruence Identify the parts of a triangle

Classify triangles

Use the Angle Sum Theorem

Identify corresponding parts

Use SSS

Use SAS

Use ASA

Use AAS

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Section 5.1 Classifying Triangles

Questions to think about:

•

Definition Characteristics

Example Nonexample

Definition Characteristics

Example Nonexample

TRIANGLE

VERTEX

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Examples…Classify each triangle by its angle and by it sides.

Triangle Classification by Angle Classification by Sides

(1.)

(2.)

(3.)

Classify Triangles

Classify by Angle Classify by Side

ACUTE All angles are acute

OBTUSE One obtuse angle

RIGHT One right angle

SCALENE No sides congruent

ISOCELES At least two sides congruent

The congruent sides are called legs and the third side

is the base

EQUILATERAL All sides are congruent

EQUIANGULAR All three angles congruent

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(4.)

(5.)

(6.)

(7.)

(8.)

Examples…using algebra

(9.) Find the measures of AB and

BC of isosceles triangle ABC if

∠A is the vertex angle.

(10.) Find the measures of XY and

YZ of isosceles triangle XYZ if

∠X is the vertex angle.

Page 5 of 13

Section 5.2 Angles of a Triangle

Questions to think about:

�

THEOREM ANGLE SUM THEOREM

5.1 The sum of the measure of the angles of a trianlge is 180.

Examples…

(11.) Find m∠T in △RST. (12.) Find the value of each variable in △DCE.

(13.) Find m∠L in △MNL if m∠M=25 and m∠N=25.

(14.) Find the value of each variable in the figure.

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(15.) Find m∠P in △MNP if m∠M=80 and m∠N=45.

(16.) Find the value of each variable △ABC.

THEOREM

5.2 The acute angles of a right triangle are

complementary.

Examples…

(17.) Find m∠A and m∠B in right triangle ABC.

(18.) Find m∠J and m∠K in right triangle JKL.

THEOREM

5.3 The measure of each angle of an equiangular triangle is 60.

Page 7 of 13

Section 5.4 Congruent Triangles

Questions to think about:

�

Definition Characteristics

Example Nonexample

Definition Characteristics

Example Nonexample

CONGRUENT TRIANGLES

CORRESPONDING PARTS

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Definition: Corresponding Parts of Corresponding Triangles are Congruent

Characteristics

Example Nonexample

Examples…

(19.) If △PQR ≅ △MLN, name the congruent angles and sides. Then draw the triangles, using arcs and slash

marks to show the congruent angles and sides.

(20.) The corresponding parts of two congruent triangles are marked on the figure. Write a congruence statement for the two triangles.

(21.) The corresponding parts of two congruent triangles are marked on the figure. Write a congruence statement for the two triangles.

CONGRUENT TRIANGLES (CPCTC)

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(22.) △RST is congruent to △XYZ. Find

the value of n.

(23.) △UVW is congruent to △GHI. If m∠V = 90 and

m∠H = 3x + 15, find the value of x.

Section 5.5 and 5.6 SSS, SAS, ASA, AAS

Questions to think about:

�

POSTULATE SSS- Side Side Side

5.1 If three sides of one triangle are congruent to three corresponding sides of another

triangle, then the triangles are congruent.

If DEAB ≅ , EFBC ≅ , and FDCA ≅

then △△△△ABC ≅≅≅≅ △△△△DEF.

Examples…

(24.) In two triangles, MLPQ ≅ , MNPR ≅ , and NLRQ ≅ . Write a congruence statement for the two triangles.

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Definition Characteristics

Example Nonexample

POSTULATE SAS- Side Angle Side

5.2 If two sides and the included angle of one triangle are congruent to the corresponding

sides and included angle of

another trianle, then the

triangles are congruent.

If MABO ≅ , ∠∠∠∠O ≅≅≅≅ ∠∠∠∠A and

ANOW ≅

then △△△△BOW ≅≅≅≅ △△△△MAN.

Examples…

(25.) Determine whether the triangles shown are congruent. If so, write a congruence statement and explain why the triangles are congruent. If not, explain why not.

(26.) Determine whether the triangles shown are congruent. If so, write a congruence statement and explain why the triangles are congruent. If not, explain why not.

INCLUDED ANGLE

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Definition Characteristics

Example Nonexample

POSTULATE ASA- Angle Side Angle

5.3 If two angles and the included

side of one triangle are

congruent to the corresponding

angles and included side of the

another triangle, then the

triangles are congurent.

Examples…

(27.) In △PQR and △KJL, ∠R ≅ ∠K, KLRQ ≅ , and ∠Q ≅ ∠L. Write a congruence statement for the two triangles.

(28.) In △DEF and △LMN, ∠D ≅ ∠N, NLDE ≅ , and ∠E ≅ ∠L. Write a congruence statement for the two triangles.

INCLUDED SIDE

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POSTULATE AAS- Angle Angle Side

5.4 If two angles and a

nonincluded side of one

triangle are congruent to the

corresponding two angles and

noninlcuded side of another

trianlge, then the triangles are

congruent.

Examples…

(29.) △ABC and △DEF each have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent AAS?

(30.) △DEF and △LMN each have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent AAS?

(31.) What other pair of angles must be marked so that two triangles are congruent by ASA?

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Examples…Determine if the triangles are congruent by SSS, SAS, AAS, ASA. If not possible to prove congruent,

write not congruent.

(32.) (33.)

(34.) (35.)