﻿ Angles in a Triangle Worksheet | Problems & Solutions

# Angles in a Triangle Worksheet

Angles in a Triangle Worksheet
• Page 1
1.
The measures of the angles of a triangle are shown below. In which case $x$ is an integer?
Case A: $x$, $x$, 4$x$
Case B: 2$x$, 3$x$, 2$x$
Case C: 6$x$, 8$x$, 3$x$
Case D: 4$x$, 2$x$, 5$x$
 a. Case D b. Case C c. Case B d. Case A

#### Solution:

Consider each case.

In case A: x, x, 4x are the measures of the angles of the triangle.

The sum of the measure of angles of a triangle is 180.
[Triangle angle sum theorem.]

So, x + x + 4x = 180
[From step 2.]

6x =180
[Simplify.]

x = 30 which is an integer.
[Solve for x.]

In other three cases the sum of the measures of angles is 7x, 17x and 11x.

7x = 180, 17x = 180, 11x = 180
[From step 2.]

x = 25.714, x = 10.588, x = 16.363 which are not integers.

2.
Find the values of $x$ and $y$.

 a. 36 and 36 b. 72 and 36 c. 72 and 72 d. 36 and 72

#### Solution:

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.
[Exterior angle theorem.]

y = 2x
[From step 1.]

The sum of the measure of the angles of a triangle is 180.
[Triangle angle-sum theorem.]

x + y + y = 180
[From step 3.]

x + 2x + 2x = 180
[Substitute y = 2x.]

5x = 180 x = 36
[Simplify.]

y = 2x 2(36) = 72
[From step 2.]

Hence the value of x is 36 and y is 72.

3.
Classify the triangle by its angle and by its sides.

 a. right scalene b. obtuse isosceles c. obtuse scalene d. right isosceles

#### Solution:

In scalene triangle no sides are congruent.

An isosceles triangle is a triangle with atleast two congruent sides.

An obtuse triangle has one angle who measure is greater than 90°.

A right triangle consists of one angle whose measure is 90°.

The given triangle consists of one right angle and two congruent sides.

Hence it is a right isosceles triangle.

4.
Find $m$$\angle$1 + $m$$\angle$2 + $m$$\angle$3 + $m$$\angle$4 + $m$$\angle$5 + $m$$\angle$6.

 a. 480 b. 540 c. 180 d. 360

#### Solution:

The measure of each exterior angle of a triangle equals the sum of its two remote interior angles.
[Exterior angle theorem.]

m1 = m4 + m6
[From step 1.]

m3 = m2 + m6
[From step 1.]

m5 = m2 + m4
[From step 1.]

The sum of the measures of the angles of a triangle is 180.
[Triangle angle sum theorem.]

m2 + m4 + m6 = 180
[From step 5.]

m1 + m2 + m3 + m4 + m5 + m6 = 180 + m1 + m3 + m5
[From step 6.]

180 + m4 + m6 + m2 + m6 + m2 + m4
[From steps 2, 3 and 4.]

180 + 180 + 180 = 540
[Simplify.]

Hence m1 +m2 + m3 + m4 + m5 + m6 = 540

5.
In traingle ABC, $m$$\angle$A = (3$x$ + 1), $m$$\angle$B = 2$x$, $m$$\angle$C = (2$x$ + 4). Find the measure of each angle.
 a. 76, 50, 54 b. 29, 27, 124 c. 79, 52, 49 d. 30, 28, 122

#### Solution:

The sum of the measures of the angles of a triangle is 180.
[Triangle angle theorem.]

(3x + 1) + 2x + 2x + 4 = 180
[From step 1.]

7x = 175 x = 25
[Simplify.]

mA = 3x + 1 3(25) + 1 = 76

mB = 2x 2(25) = 50

mC = 2x + 4 2(25) + 4 = 54

Hence the measures of angles A, B and C are 76, 50, 54.

6.
Find the measure of the angle numbered 6 in the figure.

 a. 75 b. 45 c. 55 d. 50

#### Solution:

m2 + 75 = 130
[Exterior angle theorem.]

m2 = 55

m3 = m2 = 55
[Vertical angles.]

The sum of the measures of the angles of a triangle is 180°.
[Triangle angle-sum theorem.]

EFG is a triangle, m4 + 45 + 60 = 180

m4 = 180 - 45 - 60

m4 = 75
[Simplify.]

m4 = m5 = 75
[Vertical angles.]

CDE is a triangle, m3 + m6 + m5 = 180

m6 = 180 - 75 - 55
[Substitution property.]

m6 = 50

7.
Find the values of $a$ and $b$ from the figure.

 a. 70° and 60° b. 50° and 70° c. 60° and 50° d. 70° and 50°

#### Solution:

[Exterior Angle theorem.]

mABD = 110 - 60
[Substitution property.]

mABD = 50

The sum of the measures of the angles of a triangle is 180°.
[Angle sum theorem.]

ABD is a triangle, mABD + mADB + mDAB = 180
[From step 3.]

mDAB = 180 - 60 - 50
[Substitution property.]

mDAB = 70

[Alternate interior angles]

a, CBD and ABD form a straight angle. So, a + mCBD + mABD = 180

a = 180 - 60 - 50
[Substitution property.]

a = 70

BCD is a triangle, mBCD + mBDC + mDBC = 180

b = 180 - 70 - 60
[Substitution property.]

b = 50

8.
If $\stackrel{‾}{\mathrm{AB}}$ is perpendicular to $\stackrel{‾}{\mathrm{BD}}$, find the values of $x$ and $y$.

 a. $x$ = 50° and $y$ = 30° b. $x$ = 50° and $y$ = 40° c. $x$ = 40° and $y$ = 40° d. $x$ = 40° and $y$ = 45°

#### Solution:

BCE and DCE form a linear pair.

So, mDCE + 100 = 180
[Definition of Linear Pair.]

mDCE = 180 - 100 = 80
[Substitution property.]

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

CDE is a triangle, mECD + mCDE + mDEC = 180

80 + 60 + x = 180
[Substitution property.]

x = 180 - 140 = 40

ABC is a right angled triangle. So, mBAC + mBCA = 90

mBAC = 90 - 80 = 10
[Substitution property.]

mAFB = mGFC = 130
[Vertical angles.]

AFB is a triangle, mFBA + mFAB + mAFB = 180
[From step 4.]

y = 180 - mBFA + mBAF

y = 40
[Substitution property.]

9.
Virgo is one of the constellations of Zodaic. Three principal stars of the constellation form a triangle. If $m$$\angle$S = 25 and $m$$\angle$P = 135, then find the $m$$\angle$V.

 a. 25 b. 15 c. 30 d. 20

#### Solution:

The sum of the measures of the angles of a triangle is 180°.
[Angle Sum Theorem.]

SPV is a triangle, mS + mP + mV = 180
[From step 1.]

mV = 180 - 135 - 25
[Substitution property.]

mV = 20

10.
In a triangle XYZ, if $m$$\angle$X is 20 more than $m$$\angle$Y and 25 more than $m$$\angle$Z, then find $m$$\angle$X.
 a. 55 b. 50 c. 70 d. 75

#### Solution:

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

mX + mY + mZ = 180

mY = mX - 20 and mZ = mX - 25

mX + (mX - 20) + (mX - 25) =180
[Substitution property.]

mX = 75
[On simplification.]