CHAPTER 4SECTIONS 1

AND 2Classifying Triangles

Measuring Angles in Triangles

WARM-UP1) Name a pair of consecutive interior angles. 2)  If line l is parallel to line AB, name a pair of congruent angles and state why they are congruent.  3) If line l is parallel to line AB, name a pair of supplementary angles. 4) If line AB represents the x-axis and line AC represents the y-axis, is the slope of line CB positive, negative, zero, or undefined.

A B

5

4 3 2

1l

C

10

98 7

612

11

WARM-UP1) Name a pair of consecutive interior angles.EX: <5 and <4<6 and <3

2)  If line l is parallel to line AB, name a pair of congruent angles and state why they are congruent.EX: <1 and <2 because they are corresponding angles.< 9 and <3 because they are alternate interior angles.  

A B

5

4 3 2

1l

C

10

98 7

612

11

WARM-UP3) If line l is parallel to line AB, name a pair of supplementary angles.EX: <9 and <2<5 and <4 4) If line AB represents the x-axis and line AC represents the y-axis, is the slope of line CB positive, negative, zero, or undefined.Negative  

A B

5

4 3 2

1l

C

10

98 7

612

11

VOCABULARYTriangle- A three-sided polygon

Polygon- A closed figure in a plane that is made up of segments.

Acute Triangle- All the angles are acute.

Obtuse Triangle- One angle is obtuse.

Right Triangle- One angle is right.

Hypotenuse

Leg

Leg

VOCABULARY CONT.Equlangular Triangle- An acute triangle in which all angles are congruent.

Scalene Triangle- No two sides are congruent.

Isosceles Triangle- At least two sides are congruent.

Equilateral Triangle- All the sides are congruent.

Angle Sum Theorem- The sum of the measures of the angles of a triangle is 180.

Vertex angleLeg

Base angle Base

VOCABULARY CONT.Auxiliary Line- A line or line segment added to a diagram to help in a proof.

Third Angle Theorem- If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Exterior Angle- An angle that forms a linear pair with one of the angles of the polygon.

Interior Angle- An angle inside a polygon.

Exterior angle

Interior angles

VOCABULARY CONT.Remote Interior Angles- The interior angles of the triangle not adjacent to a given exterior angle.

Exterior Angle Theorem- The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

Exterior angle

Remote Interior angles

Corollary- A statement that can be easily proven using a theorem.

Corollary- The acute angles of a right triangle are complementary.

Corollary- There can be at most one right or obtuse angle in a triangle.

Example 1: Triangle PQR is an equilateral triangle. One side measures 2x + 5 and another side measures x + 35. Find the length of each side.

Since it is an equilateral triangle all the sides are congruent.

2x + 5 = x + 35x + 5 = 35x = 5

Plug 5 in for x in to either equation.

x + 355 + 3540

Each side of the triangle is 40.

(2x + 5) (x + 35)

R Q

P

Example 2: Triangle PQR is an isosceles triangle. <P is the vertex angle, PR = x + 7 , RQ = x – 1, and QP = 3x – 5. Find x, PR, RQ, and QP.

Since it is an isosceles triangle and we know that <P is the vertex angle, PR is congruent to PQ.PR = PQx + 7 = 3x - 57 = 2x - 512 = 2x6 = x

Plug 6 in for x in the equation for PR.PR = x + 7PR = 6 + 7PR = 13 = PQ

(x + 7) (3x - 5)

R Q

P

(x - 1)

Plug 6 in for x in the equation for RQ.RQ = x – 1RQ = 6 – 1RQ = 5

Example 3: Given triangle STU with vertices S(2,3), T(4,3), and U(3,-2), use the distance formula to prove triangle STU is isosceles.

If it is isosceles two of the sides have the same length.The distance formula is d=√((x2 – x1)2 + (y2 – y1)2)

STd=√((x2 – x1)2 + (y2 – y1)2)d=√((2 – 4)2 + (3 – 3)2)d=√((-2)2 + (0)2)d=√(4 + 0)d=√(4)d= 2TUd=√((x2 – x1)2 + (y2 – y1)2)d=√((4 – 3)2 + (3 – -2)2)d=√((4 – 3)2 + (3 + 2)2)d=√((1)2 + (5)2)d=√(1 + 25)d=√(26)

SUd=√((x2 – x1)2 + (y2 – y1)2)d=√((2 – 3)2 + (3 – -2)2)d=√((2 – 3)2 + (3 + 2)2)d=√((-1)2 + (5)2)d=√(1 + 25)d=√(26)

Since TU and SU are congruent but ST is not this is an isosceles triangle.

Example 4: Given triangle STU with vertices S(2,6), T(4,-5), and U(-3,0), use the distance formula to prove triangle STU is scalene.

If it is scalene none of the sides are congruent.The distance formula is d=√((x2 – x1)2 + (y2 – y1)2)

STd=√((x2 – x1)2 + (y2 – y1)2)d=√((2 – 4)2 + (6 – -5)2)d=√((2 – 4)2 + (6 + 5)2)d=√((-2)2 + (11)2)d=√(4 + 121)d=√(125)

TUd=√((x2 – x1)2 + (y2 – y1)2)d=√((4 – -3)2 + (-5 – 0)2)d=√((4 + 3)2 + (-5 - 0)2)d=√((7)2 + (-5)2)d=√(49 + 25)d=√(74)

SUd=√((x2 – x1)2 + (y2 – y1)2)d=√((2 – -3)2 + (6 – 0)2)d=√((2 + 3)2 + (6 - 0)2)d=√((5)2 + (6)2)d=√(25 + 36)d=√(61)

Since none of the sides are congruent, this is a scalene triangle.

Example 5: A surveyor has drawn a triangle on a map. One angle measures 42 degrees and another measures 53 degrees. Find the measure of the third angle.

42

x 53

According to the angle sum theorem, all the angles in a triangle add up to 180.

180 = 42 + 53 + x180 = 95 + x85 = x

So the third angle is 85 degrees.

Example 6: A surveyor has drawn a triangle on a map. One angle measures 41 degrees and another measures 74 degrees. Find the measure of the third angle.

41

x 74

According to the angle sum theorem, all the angles in a triangle add up to 180.

180 = 41 + 74 + x180 = 115 + x65 = x

So the third angle is 65 degrees.

Example 7: Find the measure of each numbered angle in the figure if line l is parallel to line m. l

135m

5

3

2

1

60

4

m<1<1 and 135 are supplementary.180 = m<1 + 13545 = m<1

m<5<1 and <5 are alternate interior angles so they are congruent.m<1 = m<545 = m<5

m<2<5, <2, and 60 are supplementary.180 = m<5 + m<2 + 60180 = 45 + m<2 + 60180 = 105 m<275 = m<2

m<3All the angles in a triangle add up to 180.180 = m<1 + m<2 + m<3180 = 45 + 75 + m<3180 = 120 + m<360 = m<3

m<4<3 and <4 are supplementary.180 = m<3 + m<4180 = 60 + m<4120 = m<4

Example 8: Find x, y, and m<ABC

80

x y (3x – 22)

CB

A

Find xAccording to the exterior angle theorem the remote interior angles are add up to the exterior angle.

3x – 22 = 80 + x2x -22 = 802x = 102x = 51

Find yAll the angles in a triangle add up to 180.

180 = 80 + x + y180 = 80 + 51 + y180 = 131 + y49 = y

m<ABCPlug 51 in for x in the equation for m<ABC.

m<ABC = 3x – 22m<ABC = 3(51) – 22m<ABC = 153 – 22m<ABC = 131