Chapter 4

Equilateral Triangle

A triangle with three congruent sides.

transversal

Alternate Interior Angles Theorem: If two parallel lines are cut by a ___, then the alternate interior angles are congruent.

CPCTC

An abbreviation for "Corresponding Parts of Congruent Triangles are Congruent," which can be used as a justification in a proof after two triangles are proven congruent.

Sometimes

An acute triangle is (A, S, or N) an equiangular triangle.

Exterior Angle of Triangle

An angle that forms a linear pair with an interior angle of the triangle.

Right Angle

An angle with measure of 90 degrees.

Always

An equiangular triangle is (A, S, or N) an acute triangle.

Always

An equilateral triangle is (A, S, or N) an equiangular triangle.

Sometimes

An obtuse triangle is (A, S, or N) a scalene triangle.

Never

An obtuse triangle is (A, S, or N) an equilateral triangle.

AAS

Are the triangles congruent? If yes, why?

ASA

Are the triangles congruent? If yes, why?

HL

Are the triangles congruent? If yes, why?

SAS

Are the triangles congruent? If yes, why?

SSS

Are the triangles congruent? If yes, why?

NO

Can you use AAA to prove congruent triangles?

NO

Can you use ASS to prove congruent triangles?

NO

Can you use SSA to prove congruent triangles?

Isosceles

Classify the triangle by side lengths.

90

Definition of Perpendicular: Perpendicular lines form ___ degree angles.

right

Definitional of Right Triangle: Right Triangles have ___ angles.

c + 34 + 120 = 180 c + 154 = 180 c = 26 26 degrees

Find the measure of angle C.

y + 42 + 42 = 180 y + 84 = 180 y = 96

Find the measure of angle Y.

x = 66 y = 180 - 66 - 66 y = 48

Find x and y.

3x = 60 x = 20 4y = 60 y = 15 5z = 60 z = 12

Find x, y and z.

(4x + 2) + (2x - 9) = 5x + 13 6x - 7 = 5x + 13 x - 7 = 13 x = 20

Find x.

25 + (x + 15) = 3x - 10 x + 40 = 3x - 10 40 = 2x - 10 50 = 2x 25 = x or x = 25

Find x.

2x + 3x = 100 5x = 100 x = 20

Find x.

4x = 60 x = 15

Find x.

5y - 6 = 4y + 12 y - 6 = 12 y = 18

Find y.

Converse

Formed by switching conclusion and hypothesis of conditional.

2

How many base angles does an isosceles triangle have?

3

How many interior angles does a triangle have?

1

How many vertex angles does an isosceles triangle have?

X

If Triangle ABC is congruent to Triangle WXY, then Angle B is congruent to Angle ___.

WY

If Triangle ABC is congruent to Triangle WXY, then Segment AC is congruent to Segment ___.

equiangular

If each interior angle of a triangle has a measure of 60 degrees, then the triangle is ___.

1 and 2

If segment AB is parallel to segment CD, then these two angles are congruent.

3 and 4

If segment AD is parallel to segment BC, then these two angles are congruent.

alternate

If two parallel lines are cut by a transversal, then the ___ interior angles are congruent.

opposite

Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles ___ those sides are congruent.

Angle X

Name the angle included by segments XY and XZ.

Angle Z

Name the angle included by segments XZ and YZ.

Angle Y

Name the angle included by segments YZ and XY.

HL, ASA, SAS, AAS, and SSS

Name the five postulates and theorems you can use to prove congruent triangles.

segment AB

Name the side included by angles A and B.

Segment AC

Name the side included by angles A and C.

Segment BC

Name the side included by angles B and C.

Leg

One of the two congruent sides of an isosceles triangle.

Reflexive Property

Reason an angle or segment is congruent to itself.

congruent

Right Angle Congruence Thm: Right angles are ____.

90

The acute angle measures of a right triangle have a sum of ___ degrees.

Vertex Angle

The angle formed by the legs of an isosceles triangle.

Included Angle

The angle formed by two adjacent sides of a triangle.

obtuse and scalene

The interior angle measures of a triangle are 16, 100, and 64 degrees. The triangle is ___ and ___.

180

The interior angle sum of a triangle is ___ degrees.

smaller angle: x larger angle: 4x x + 4x = 90 x = 18 18 degrees

The measure of one acute angle in a right triangle is 4 times the measure of the other acute angle. What is the measure of the smaller angle?

smaller angle: x larger angle: 5x x + 5x = 90 x = 15 15 degrees

The measure of one acute angle in a right triangle is 5 times the measure of the other acute angle. What is the measure of the smaller angle?

smaller angle: x larger angle: 8x x + 8x = 90 x = 10 10 degrees

The measure of one acute angle in a right triangle is 8 times the measure of the other acute angle. What is the measure of the smaller angle?

Multiplier = 180 / (1 + 2 + 3) = 180 / 6 = 30 Smallest Angle = 1(30) = 30 degrees Largest Angle = 3(30) = 90 degrees

The ratio of the measures of three angles of a triangle is 1:2:3. Find the measures of the smallest and largest angles.

AAS

Why are Triangles PRS and QRS congruent?

Corresponding Angles Postulate

_____ _____ Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Equiangular Triangle

A triangle with three congruent angles.

Midpoint

A point that divides a segment into two congruent segments

Corollary

A theorem whose proof follows directly from another theorem.

Isosceles Triangle

A triangle with at least two congruent sides.

Scalene Triangle

A triangle with no congruent sides.

Obtuse Triangle

A triangle with one obtuse angle.

Right Triangle

A triangle with one right angle.

Acute Triangle

A triangle with three acute angles.

Multiplier = 180 / (1 + 4 + 7) = 180 / 12 = 15 Smallest Angle = 1(15) = 15 degrees Largest Angle = 7(15) = 105 degrees

The ratio of the measures of three angles of a triangle is 1:4:7. Find the measures of the smallest and largest angles.

Multiplier = 180 / (2 + 3 + 4) = 180 / 9 = 20 Smallest Angle = 2(20) = 40 degrees Largest Angle = 4(20) = 80 degrees

The ratio of the measures of three angles of a triangle is 2:3:4. Find the measures of the smallest and largest angles.

Base

The side opposite of the vertex angle.

x = 7 (Set any two sides equal to each other>)

The triangle is equilateral. Find x.

Base Angles

The two angles that have the base as the side.

Remote Interior Angle

The two nonadjacent interior angles corresponding to each exterior angle of a triangle.

2x + (5x + 3) + (5x + 3) = 180 12x + 6 = 180 12x = 174 x = 14.5

The vertex angle of an isosceles triangle measures (2x) degrees, and one of the base angles measures (5x + 3) degrees. Find x.

2x + 50 + 50 = 180 2x + 100 = 180 2x = 80 x = 40

The vertex angle of an isosceles triangle measures (2x) degrees, and one of the base angles measures 50 degrees. Find x.

5x + 5x + 40 = 180 x = 14

The vertex angle of an isosceles triangle measures 40 degrees, and one of the base angles measures (5x) degrees. Find x.

Bisect

To divide into two congruent parts.

True

True or False. Every equilateral triangle is also isosceles.

False

True or False. Every isosceles triangle is also equilateral.

Congruent Triangles

Two triangles whose corresponding sides and angles are congruent.

congruent

Vertical Angles Theorem: Vertical angles are ___.