Unit 5Congruent Triangles

Section 1Angles of Triangles

Classifying by SidesScalene

Isosceles

Equilateral

Classifying by AnglesAcute

Right

Obtuse

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Example 1.1Classify the triangular shape of thesupport beams in the diagram by its sides and its angles.

Example 1.2Draw triangle PQO and classify it by its sides. Then determine if the triangle is a right triangle.P(-1, 2) Q(6, 3) O(0, 0)

Example 1.3

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With your POD:Triangle ABC has vertices A(0, 0), B(3, 3), and C(-3, 3). Classify it by its sides. Then determine if it is a right triangle.

Interior VS Exterior Angles

Interior Angles:

Exterior Angles:

Theorem: Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180o.

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Proof of Triangle Sum TheoremGiven triangle ABC. Prove the angle sum is 180o.Draw an auxiliary line through B and parallel to side AC.

Theorem: Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Example 1.4Find m<JKM

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Corollary to the Triangle Sum TheoremThe acute angles of a right triangle are complementary.

Example 1.5Find the measure of each acute angle.

Example 1.6Find the measure of each interior angle of triangle ABC, where m≮A=x, m≮B=2x, and m≮C=3x

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Section 2Congruent Polygons

Congruent Figures: have exactly the same __________________.

All corresponding parts are _________________

Corresponding Parts: a pair of ________________________ that

have the same relative position in two congruent or similar figures.

Congruence Statements*A way of writing that two figures are congruent*Always list corresponding vertices in the same order*Not unique- more than one possible congruence statement

Writing Congruence Statements

Corresponding Angles:

Corresponding Sides:

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Example 2.1Write a congruence statement and identify all sets of corresponding parts.

Example 2.2Write a congruence statement and identify all pairs of corresponding parts.

Example 2.3DEFG≌SPQR. Find the value of x and the value of y.

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Example 2.4If you divide the wall into orange and blue sections along JK, will the sections of the wall be the same size and shape? Explain.

Example 2.5ABGH≌CDEFIdentify all pairs of congruent corresponding parts.

Find the value of x and find the m∠H.

Example 2.6Show that ∆PTS≌∆RTQ

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Theorem: Properties of Triangle CongruenceTriangle congruence is reflexive, symmetric, and transitive.

Reflexive:

Symmetric:

Transitive:

Theorem: Third Angles TheoremIf two angles of one triangle are congruent to two angles of another

triangle, then __________________________________________

Example 2.7Find the m∠BDC

Example 2.8Prove that ∆ACD≌∆CAB

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Example 2.9∆XYZ≌∆LMN and ∆NLM≌∆HJK, if m∠Y=48º and m∠X=73º, find m∠H.

Example 2.10Find the m∠DCN

What additional information is needed to know that ∆NDC≌∆NSR?

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Section 3SAS and HL Congruence

Theorem: Side-Angle-Side (SAS) Congruence TheoremIf two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Identifying Included Angles

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Example 3.1Given: BC≌DA and BC||ADProve: ∆ABC≌∆CDA

Example 3.2In the diagram, QS and RP pass through the center M of the circle. What can you conclude about ∆MRS and ∆MPQ?

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Example 3.3In the diagram, ABCD is a square with four congruent sides and four right angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, RT⊥SU and SV≌VU.

Right Triangles

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What is a right triangle?

Legs: Sides adjacent tot he right angleHypotenuse: Side opposite the right angle

Theorem: Hypotenuse-Leg (HL) Congruence TheoremIf the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two right triangles are congruent.

Example 3.4Given: WY≌XZ, WZ⊥ZY, XY⊥ZYProve: ∆WXY≌∆XZY

Example 3.5

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You are making a canvas sign to hang on the triangular wall over the door to the barn shown in the picture. You think you can use two identical triangular sheets of canvas. You know that RP⊥QS and PQ≌PS. What postulate or theorem can you use to conclude that ∆PQR≌∆PSR?

Section 4Equilateral and Isosceles Triangles

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Isosceles Triangles

Theorem: Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite

them are ___________________.

Theorem: converse of the Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite

them are ___________________.

Example 4.1In ∆DEF, DE≌DF. Name two congruent angles.Example 4.2If HG≌HK, then ∠____≌∠____.

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If ∠KHJ≌∠KJH, then ____≌____.

Corollary: Corollary to the Base Angles Theorem

If a triangle is _________________, then it is ________________

Corollary: Corollary to the Converse of the Base Angles

Theorem

If a triangle is _________________, then it is ________________

Example 4.3Find the measures of ∠P, ∠Q, and ∠R.

Example 4.4Find ST in the triangle.

Is it possible for an equilateral to have an angle measure other than 60º?

Example 4.5Find the values of x and y.

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Example 4.6Find the values of x and y.

Section 5SSS Congruence

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Theorem: Side-Side-Side (SSS) Congruence TheoremIf three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

Example 5.1Given: KL≌NL and KM≌NMProve: ∆KLM≌∆NLM

Example 5.2Decide whether the congruence statement is true. Explain your

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reasoning.1. △DFG ≅ △HJK 2. △ACB ≅ △CAD 3. △QPT ≅ △RST

Example 5.3Explain why the bench with the diagonal support is stable, while the one without the support can collapse.

ConstructUse a protractor and a straight-edge to construct a congruent triangle to ∆ABC using the SSS congruence postulate.

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Section 6ASA and AAS Congruence

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Theorem: Angle-Side-Angle (ASA) Congruence TheoremIf two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

Theorem: Angle-Angle-Side (AAS) Congruence TheoremIf two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

Example 6.1Can the triangles be proven congruent with the information given in the picture? If so, tell what postulate or theorem you would use.

ProofProve the Angle-Angle-Side Congruence Theorem.

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Given: ∠A≌∠D, ∠C≌F, BC≌EFProve: ∆ABC≌∆DEF

Example 6.2Can the triangles be proven congruent using the information given? If so, which theorem did you use?

Example 6.3Are the triangles congruent?

Example 6.4

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What postulate or theorem can you use to prove that ∆RST≌∆VUT?

Example 6.5Prove that the two triangles are congruent.

Example 6.6

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Prove that the two triangles are congruent.

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