4.1:4.1: Apply Triangle Sum Apply Triangle Sum PropertiesProperties

AimAim:: To classify triangles and find To classify triangles and find measures of their angles. measures of their angles.

What is a triangle? What is a triangle?

Classifying Triangles by SidesClassifying Triangles by Sides

Scalene TriangleScalene Triangle::– Has no congruent sides.Has no congruent sides.

ac

Isosceles TriangleIsosceles Triangle:: - Has at least two congruent sides.- Has at least two congruent sides.

Equilateral TriangleEquilateral Triangle:: - Has three congruent sides.- Has three congruent sides.

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Classifying Triangles by AnglesClassifying Triangles by Angles

Acute TriangleAcute Triangle::- Has three acute angles.- Has three acute angles.

Right TriangleRight Triangle::- Has one right angle.- Has one right angle.

Obtuse TriangleObtuse Triangle::- Has one obtuse angle.- Has one obtuse angle.

Equiangular TriangleEquiangular Triangle::- Has three congruent angles.- Has three congruent angles.

Graph Triangle Graph Triangle ABCABC Graph Graph AA(-5, 4), (-5, 4), BB((2, 6), and 2, 6), and CC(4, -1).(4, -1).

212

212 yyxxd

•

••A

B

C

Classify the triangle by its sides using the Classify the triangle by its sides using the distance distance formulaformula. .

This is an Isosceles triangle. This is an Isosceles triangle.

106

53

53

AC

BC

AB

212

212 yyxxd

How can we check if any of the angles How can we check if any of the angles are right angles?are right angles?

Using the coordinates Using the coordinates AA(-5, 4), (-5, 4), BB((2, 6), and 2, 6), and CC(4, -(4, -1). Find the 1). Find the slopeslope of each line. of each line.

9

52

77

2

AC

BC

AB What type of triangle is this considered to be?

Right Isosceles Triangle

Mini-ActivityMini-Activity

Move your desk to the person next to Move your desk to the person next to you.you.

Draw 1 straight line (use your ID card Draw 1 straight line (use your ID card for a straight edge). for a straight edge).

Take the Triangle give to you and Take the Triangle give to you and tear off each angle. tear off each angle.

TheoremsTheorems Triangle Sum TheoremTriangle Sum Theorem::

– The sum of the measures of the interior The sum of the measures of the interior angles of a triangle is 180°.angles of a triangle is 180°.

Exterior Angle TheoremExterior Angle Theorem::– The measure of an exterior angle of a triangle The measure of an exterior angle of a triangle

is equal to the sum of the measures of the is equal to the sum of the measures of the two nonadjacent interior angles.two nonadjacent interior angles.

Corollary to the Triangle Sum TheoremCorollary to the Triangle Sum Theorem::– The acute angles of a right triangle are The acute angles of a right triangle are

complementary.complementary.

4.2:4.2: Apply Congruence and TrianglesApply Congruence and TrianglesDate:Date: 11/211/2

AimAim:: To identify congruent triangles. To identify congruent triangles. Do NowDo Now::Find m∠JKM.

Find an angle measure

STEP 1 Write and solve an equation to find the value of x.

Apply the Exterior Angle Theorem.(2x – 5)° = 70° + x°

Solve for x.x = 75

STEP 2Substitute 75 for x in 2x – 5 to find m∠JKM.

2x – 5 = 2 75 – 5 = 145

Find m∠JKM.

The measure of ∠JKM is 145°.ANSWER

Find angle measures from a verbal description

Use the corollary to set up and solve an equation.

Corollary to the Triangle Sum Theoremx° + 2x° = 90°

Solve for x.x = 30

So, the measures of the acute angles are 30° and 2(30°) = 60° .

ANSWER

Find the measure of 1 in the diagram shown.3.

The measure of ∠1 in the diagram is 65°.ANSWER

Third Angle TheoremThird Angle Theorem

If two angles of one triangle are If two angles of one triangle are congruent to two angles of another congruent to two angles of another triangle, then the third angles are triangle, then the third angles are also congruent. also congruent.

4.2:4.2: Apply Congruence and Apply Congruence and TrianglesTriangles

Date:Date: 11/8/1011/8/10

AimAim:: To identify congruent To identify congruent triangles. triangles.

Do NowDo Now: : Take out Homework.Take out Homework.

4.3:4.3: Prove Triangles Congruent by Prove Triangles Congruent by SSSSSS

Date:Date: 11/9/1011/9/10

AimAim:: To use the side lengths to To use the side lengths to prove triangles are congruent. prove triangles are congruent.

PostulatePostulate

Side-Side-Side (SSS) Congruence Side-Side-Side (SSS) Congruence PostulatePostulate– If three sides of one triangle are If three sides of one triangle are

congruent to three sides of another congruent to three sides of another triangle, then the two triangles are triangle, then the two triangles are congruent.congruent.

4.4:4.4: Prove Triangles Congruent by Prove Triangles Congruent by SAS and HLSAS and HL

Date:Date: 11/10/1011/10/10

AimAim:: To use sides and angle To use sides and angle lengths to prove congruence. lengths to prove congruence.

Do NowDo Now::

Page 243 Page 243 #’s (9 – 14, 20 – 22, #’s (9 – 14, 20 – 22, 34)34)

Side-Angle-Side Congruence Postulate Side-Angle-Side Congruence Postulate (SAS)(SAS)

If two sides and the included angle of If two sides and the included angle of one triangle are congruent to two one triangle are congruent to two sides and the included angle of a sides and the included angle of a second triangle, then the two second triangle, then the two triangles are congruent. triangles are congruent.

4.4:4.4: Prove Triangles Congruent by Prove Triangles Congruent by SAS and HLSAS and HL

Date:Date: 11/12/1011/12/10

AimAim:: To use sides and angle To use sides and angle lengths to prove congruence. lengths to prove congruence.

Hypotenuse-Leg Congruence Theorem Hypotenuse-Leg Congruence Theorem (HL)(HL)

If the hypotenuse and a leg of a right If the hypotenuse and a leg of a right triangle are congruent to the triangle are congruent to the hypotenuse and a leg of a second hypotenuse and a leg of a second right triangle, then the two triangles right triangle, then the two triangles are congruent. are congruent.

4.5:4.5: Prove Triangles Congruent by Prove Triangles Congruent by ASA and AASASA and AAS

Date:Date: 11/15/1011/15/10

AimAim:: To use two more methods to To use two more methods to prove congruence (ASA and AAS). prove congruence (ASA and AAS).

Angle-Side-Angle Congruence Angle-Side-Angle Congruence Postulate (ASA)Postulate (ASA)

If two angles and the included side of If two angles and the included side of one triangle are congruent to two one triangle are congruent to two angles and the included side of a angles and the included side of a second triangle, then the two second triangle, then the two triangles are congruent. triangles are congruent.

Angle-Angle-Side Congruence Angle-Angle-Side Congruence Theorem (AAS)Theorem (AAS)

If two angles and a non-included side If two angles and a non-included side of one triangle are congruent to the of one triangle are congruent to the two angles and the corresponding two angles and the corresponding non-included side of a second non-included side of a second triangle, then the two triangle are triangle, then the two triangle are congruent. congruent.

4.6:4.6: Use Congruent TrianglesUse Congruent Triangles

Date:Date: 12/1/0912/1/09

AimAim:: To use congruent triangles to To use congruent triangles to prove corresponding parts prove corresponding parts congruent. congruent.

4.6:4.6: Use Congruent TrianglesUse Congruent Triangles

Date:Date: 12/2/0912/2/09

AimAim:: To use congruent triangles to To use congruent triangles to prove corresponding parts prove corresponding parts congruent. congruent.

4.7:4.7: Use Isosceles and Equilateral Use Isosceles and Equilateral TrianglesTriangles

Date:Date: 12/3/0912/3/09

AimAim: : To use theorems about To use theorems about isosceles and equilateral triangles. isosceles and equilateral triangles.

Base Angles TheoremBase Angles Theorem

If two sides of a triangle are If two sides of a triangle are congruent, then the angels opposite congruent, then the angels opposite them are congruent.them are congruent.

Converse of Base Angles TheoremConverse of Base Angles Theorem

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

CorollariesCorollaries

Corollary to the Base Angle TheoremCorollary to the Base Angle Theorem::– If a triangle is equilateral, then it is If a triangle is equilateral, then it is

equiangular.equiangular.

Corollary to the Converse of the Base Corollary to the Converse of the Base Angles TheoremAngles Theorem::– If a triangle is equiangular, then it is If a triangle is equiangular, then it is

equilateral. equilateral.