GEOMETRY

2-1 Triangles

Warm UpClassify each angle as acute, obtuse, or right.

1. 2.

3.

4. If the perimeter is 47, find x and the lengths of the three sides.

right acute

x = 5; 8; 16; 23

obtuse

GEOMETRY

2-1 Triangles

In the figure, n is a whole number. What is the smallest possible value for n?

A sewing club is making a quilt consisting of 25 squares with each side of the square measuring 30 centimeters. If the quilt has five rows and five columns, what is the perimeter of the quilt?

1.

2.

GEOMETRY

2-1 Triangles

In the figure, n is a whole number. What is the smallest possible value for n?

1.

GEOMETRY

2-1 Triangles

A sewing club is making a quilt consisting of 25 squares with each side of the square measuring 30 centimeters. If the quilt has five rows and five columns, what is the perimeter of the quilt?

2.

GEOMETRY

2-1 Triangles

Classify triangles by their angle measures and side lengths.

Use triangle classification to find angle measures and side lengths.

Objectives

GEOMETRY

2-1 Triangles

acute triangleequiangular triangleright triangleobtuse triangleequilateral triangleisosceles trianglescalene triangle

Vocabulary

GEOMETRY

2-1 Triangles

12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.

15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.

California Standards

GEOMETRY

2-1 Triangles

Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths.

GEOMETRY

2-1 Triangles

B

AC

AB, BC, and AC are the sides of ABC.

A, B, C are the triangle's vertices.

GEOMETRY

2-1 Triangles

Acute Triangle

Three acute angles

Triangle Classification By Angle Measures

GEOMETRY

2-1 Triangles

Equilateral (Equiangular) Triangle

Three congruent acute angles

Triangle Classification By Angle Measures

GEOMETRY

2-1 Triangles

Right Triangle

One right angle

Triangle Classification By Angle Measures

GEOMETRY

2-1 Triangles

Obtuse Triangle

One obtuse angle

Triangle Classification By Angle Measures

GEOMETRY

2-1 Triangles

Classify FHG by its angle measures.

Teach! Example 1

EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. So mFHG = 60°.

FHG is an equilateral (Equiangular) triangle by definition.

GEOMETRY

2-1 Triangles

Equilateral Triangle

Three congruent sides

Triangle Classification By Side Lengths

GEOMETRY

2-1 Triangles

Isosceles Triangle

At least two congruent sides

Triangle Classification By Side Lengths

GEOMETRY

2-1 Triangles

Scalene Triangle

No congruent sides

Triangle Classification By Side Lengths

GEOMETRY

2-1 Triangles

Remember!When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.

GEOMETRY

2-1 Triangles

Classify EHF by its side lengths.

Example 1: Classifying Triangles by Side Lengths

From the figure, . So HF = 10, and EHF is isosceles.

GEOMETRY

2-1 Triangles

Classify EHG by its side lengths.

TEACH! : Classifying the Triangle by Side Lengths

By the Segment Addition Postulate, EG = EF + FG = 10 + 4 = 14. Since no sides are congruent, EHG is scalene.

GEOMETRY

2-1 Triangles

Classify ACD by its side lengths.

TEACH! Example 3

From the figure, . So AC = 15, and ACD is scalene and probably obtuse.

GEOMETRY

2-1 Triangles

Find the side lengths of JKL.

Example 3: Using Triangle Classification

Step 1 Find the value of x.

Given.

JK = KL Def. of segs.

4x – 10.7 = 2x + 6.3Substitute (4x – 10.7) for JK and (2x + 6.3) for KL.

2x = 17.0

x = 8.5

Add 10.7 and subtract 2x from both sides.

Divide both sides by 2.

GEOMETRY

2-1 Triangles

Find the side lengths of JKL.

Example 3 Continued

Step 2 Substitute 8.5 into the expressions to find the side lengths.

JK = 4x – 10.7

= 4(8.5) – 10.7 = 23.3

KL = 2x + 6.3

= 2(8.5) + 6.3 = 23.3JL = 5x + 2

= 5(8.5) + 2 = 44.5

GEOMETRY

2-1 Triangles

Find the side lengths of equilateral FGH.

TEACH! Example 3

Step 1 Find the value of y.

Given.

FG = GH = FH Def. of segs.

3y – 4 = 2y + 3Substitute (3y – 4) for FG and (2y + 3) for GH.

y = 7 Add 4 and subtract 2y from both sides.

GEOMETRY

2-1 Triangles

Find the side lengths of equilateral FGH.

TEACH! Example 3 Continued

Step 2 Substitute 7 into the expressions to find the side lengths.

FG = 3y – 4

= 3(7) – 4 = 17

GH = 2y + 3

= 2(7) + 3 = 17FH = 5y – 18

= 5(7) – 18 = 17

GEOMETRY

2-1 Triangles

The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.

P = 3(18)

P = 54 ft

A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?

Example 4: Application

GEOMETRY

2-1 Triangles

A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?

Example 4: Application Continued

To find the number of triangles that can be made from 420 feet of steel beam, divide 420 by the amount of steel needed for one triangle.

420 54 = 7 triangles 7 9

There is not enough steel to complete an eighth triangle. So the steel mill can make 7 triangles from a 420 ft. piece of steel beam.

GEOMETRY

2-1 Triangles

The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.

P = 3(7)

P = 21 in.

Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.

TEACH! Example 4

GEOMETRY

2-1 Triangles

To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.

100 7 = 14 triangles 2 7

There is not enough steel to complete a fifteenth triangle. So the manufacturer can make 14 triangles from a 100 in. piece of steel.

Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.

TEACH! Example 4 Continued

GEOMETRY

2-1 Triangles

The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.

P = 3(10)

P = 30 in.

Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.

TEACH! Example 5

GEOMETRY

2-1 Triangles

To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.

100 10 = 10 triangles

The manufacturer can make 10 triangles from a 100 in. piece of steel.

Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.

TEACH! Example 5 Continued

GEOMETRY

2-1 Triangles

Lesson Quiz

Classify each triangle by its angles and sides.

1. MNQ

2. NQP

3. MNP

4. Find the side lengths of the triangle.

acute; equilateral

obtuse; scalene

acute; scalene

29; 29; 23

GEOMETRY

2-1 Triangles

GEOMETRY

2-1 Triangles

150°

73°

1; Parallel Post.

Warm Up

1. Find the measure of exterior DBA of BCD, if mDBC = 30°, mC= 70°, and mD = 80°.

2. What is the complement of an angle with measure 17°?

3. How many lines can be drawn through N parallel to MP? Why?

GEOMETRY

2-1 Triangles

Find the measures of interior and exterior angles of triangles.

Apply theorems about the interior and exterior angles of triangles.

Objectives

GEOMETRY

2-1 Triangles

auxiliary linecorollaryinteriorexteriorinterior angleexterior angleremote interior angle

Vocabulary

GEOMETRY

2-1 Triangles

GEOMETRY

2-1 Triangles

An auxiliary line is a line that is added to a figure to aid in a proof.

An auxiliary line used in the Triangle Sum

Theorem

GEOMETRY

2-1 Triangles

After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mXYZ.

Example 6A: Application

mXYZ + mYZX + mZXY = 180° Sum. Thm

mXYZ + 40 + 62 = 180Substitute 40 for mYZX and 62 for mZXY.

mXYZ + 102 = 180 Simplify.

mXYZ = 78° Subtract 102 from both sides.

GEOMETRY

2-1 Triangles

Use the diagram to find mMJK.

TEACH! Example 6

mMJK + mJKM + mKMJ = 180° Sum. Thm

mMJK + 104 + 44= 180 Substitute 104 for mJKM and 44 for mKMJ.

mMJK + 148 = 180 Simplify.

mMJK = 32° Subtract 148 from both sides.

GEOMETRY

2-1 Triangles

A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

GEOMETRY

2-1 Triangles

One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?

Example 7: Finding Angle Measures in Right Triangles

mA + mB = 90°

2x + mB = 90 Substitute 2x for mA.

mB = (90 – 2x)° Subtract 2x from both sides.

Let the acute angles be A and B, with mA = 2x°.

Acute s of rt. are comp.

GEOMETRY

2-1 Triangles

The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?

TEACH! Example 7a

mA + mB = 90°

63.7 + mB = 90 Substitute 63.7 for mA.

mB = 26.3° Subtract 63.7 from both sides.

Let the acute angles be A and B, with mA = 63.7°.

Acute s of rt. are comp.

GEOMETRY

2-1 Triangles

The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?

TEACH! Example 7b

mA + mB = 90°

x + mB = 90 Substitute x for mA.

mB = (90 – x)° Subtract x from both sides.

Let the acute angles be A and B, with mA = x°.

Acute s of rt. are comp.

GEOMETRY

2-1 Triangles

The measure of one of the acute angles in a

right triangle is 48 . What is the measure of

the other acute angle?

TEACH! Example 7c

mA + mB = 90° Acute s of rt. are comp.

2° 5

Let the acute angles be A and B, with mA = 48 . 2° 5

Subtract 48 from both sides. 2 5

Substitute 48 for mA. 2 548 + mB = 90

2 5

mB = 41 3° 5

GEOMETRY

2-1 Triangles

The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.

Interior

Exterior

GEOMETRY

2-1 Triangles

An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side.

Interior

Exterior

4 is an exterior angle.

3 is an interior angle.

GEOMETRY

2-1 Triangles

Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.

Interior

Exterior

3 is an interior angle.

4 is an exterior angle.

The remote interior angles of 4 are 1 and 2.

GEOMETRY

2-1 Triangles

Corollary: The measure of an exterior angle of a triangle is greater than the measure of either of its remote angles. 4 1 and 4 2m m m m

GEOMETRY

2-1 Triangles

Find mB.

Example 8: Applying the Exterior Angle Theorem

mA + mB = mBCD Ext. Thm.

15 + 2x + 3 = 5x – 60 Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.

2x + 18 = 5x – 60 Simplify.

78 = 3xSubtract 2x and add 60 to both sides.

26 = x Divide by 3.

mB = 2x + 3 = 2(26) + 3 = 55°

GEOMETRY

2-1 Triangles

Find mACD.

TEACH! Example 8

mACD = mA + mB Ext. Thm.

6z – 9 = 2z + 1 + 90 Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB.

6z – 9 = 2z + 91 Simplify.

4z = 100Subtract 2z and add 9 to both sides.

z = 25 Divide by 4.

mACD = 6z – 9 = 6(25) – 9 = 141°

GEOMETRY

2-1 Triangles

GEOMETRY

2-1 Triangles

Find mK and mJ.

Example 9: Applying the Third Angles Theorem

K J

mK = mJ

4y2 = 6y2 – 40

–2y2 = –40

y2 = 20

So mK = 4y2 = 4(20) = 80°.

Since mJ = mK, mJ = 80°.

Third s Thm.

Def. of s.

Substitute 4y2 for mK and 6y2 – 40 for mJ.

Subtract 6y2 from both sides.

Divide both sides by -2.

GEOMETRY

2-1 Triangles

TEACH! Example 9

Find mP and mT.

P T

mP = mT

2x2 = 4x2 – 32

–2x2 = –32

x2 = 16

So mP = 2x2 = 2(16) = 32°.

Since mP = mT, mT = 32°.

Third s Thm.

Def. of s.

Substitute 2x2 for mP and 4x2 – 32 for mT.

Subtract 4x2 from both sides.

Divide both sides by -2.

GEOMETRY

2-1 Triangles

Lesson Quiz: Part I

1. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle?

2. Find mABD. 3. Find mN and mP.

124° 75°; 75°

2 3

33 °1 3

GEOMETRY

2-1 Triangles

Lesson Quiz: Part II

4. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store?

30°