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, 4x
Case B: 2x, 3x, 2x
Case C: 6x, 8x, 3x
Case D: 4x, 2x, 5x
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.


Correct answer : (4)
 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.


Correct answer : (4)
 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.


Correct answer : (4)
 4.  
Find m1 + m2 + m3 + m4 + m5 + m6.

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


Correct answer : (2)
 5.  
In traingle ABC, mA = (3x + 1), mB = 2x, mC = (2x + 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.


Correct answer : (1)
 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


Correct answer : (4)
 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:

mABD + mADB = 110
[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

mCBD = mADB = 60
[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


Correct answer : (4)
 8.  
If AB is perpendicular to 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.]


Correct answer : (3)
 9.  
Virgo is one of the constellations of Zodaic. Three principal stars of the constellation form a triangle. If mS = 25 and mP = 135, then find the mV.

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


Correct answer : (4)
 10.  
In a triangle XYZ, if mX is 20 more than mY and 25 more than mZ, then find mX.
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.]


Correct answer : (4)

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