# Isosceles Triangle Worksheet

Isosceles Triangle Worksheet
• Page 1
1.
If the perimeter of an isosceles triangle is 12 times its base, then what is the ratio of the length of its leg to base?
 a. 4 : 1 b. 2 : 11 c. 1 : 4 d. 11 : 2

#### Solution:

Let x be the length of the base and y be the length of two equal sides.

x + 2 y = 12x

11x = 2y

yx = 11 / 2

Ratio of the length of leg to base = 11 : 2

2.
$m$$\angle$ABC = 57. Find the measure of $\angle$ACB.

 a. 20 b. 19 c. 28 d. 58

#### Solution:

mACB = mDBC
[BD = DC.]

[Exterior angles.]

mBAD = 180 - (mABC + mACB)
[Angles in the same triangle.]

[mABC = 57.]

mADB = 1 / 2 × (180 - (114 - mACB)
[Step 3.]

mADB = 1 / 2 × (57 + mACB)
[Step 3.]

2mDCB = 1 / 2 × (57 + mACB)
[Steps 3 and 7.]

2mACB = 1 / 2 × (57 + mACB)
[DCB = ACB.]

mACB = 19
[Solve.]

3.
ΔABC is an isosceles triangle and $x$ = 40. Select the correct statement/statements.
1. Base is longer than the legs.
2. $m$$\angle$ ACD = 110

 a. 2 only b. 1 only c. both 1 and 2 d. both are wrong

#### Solution:

[In ΔABC, AB and AC are legs.]

mBAC + mABC + mACB = 180
[Sum of the measures of the angles in a triangle.]

mABC = 1 / 2(180 - 40) = 70
[mABC = mACB.]

mBAC < mABC
[Step 3.]

BC < AC
[Side opposite to larger angle will be larger.]

Base is shorter than legs.
[Step 5.]

mACD = 180 - mACB
[Supplementary angles.]

= 180 - 70 = 110

Statement 2 only is correct.

4.
The angles 1 and 2 are in the ratio 3 : 4. Find $m$$\angle$3.

 a. 54.435 b. 48.435 c. 51.435 d. 56.435

#### Solution:

m1 : m2 = 3 : 4
[Given.]

m1 + m2 = 180
[Supplementary angles.]

Let m1 = 3x, then m2 = 4x
[Angles are in the ratio 3 : 4.]

3x + 4x = 180
[Step 2.]

x = 25.71
[Solve.]

m1 = 3x = 77.13

m3 = 1 / 2(180 - m1)
[m3 = m4.]

= 51.435

5.
Which of the following is/are true?
1. If AO = OD, then ABDC will be a rhombus
2. If AD = BC, then ABDC will be a square
3. $\stackrel{‾}{\mathrm{BC}}$ is always the perpendicular bisector of $\stackrel{‾}{\mathrm{AD}}$
4. $\stackrel{‾}{\mathrm{AD}}$ is always the perpendicular bisector of $\stackrel{‾}{\mathrm{BC}}$

 a. All are correct b. 1, 2 & 4 only c. 1, 2 & 3 only d. 1 & 2 only

#### Solution:

In ΔABC, AB = AC, AO ^ BC, OB = OC
[Isosceles triangle theorem.]

AB = AC B = C.

When AO = OD, AB = AC = BD = CD; ABDC becomes a rhombus.
[All the sides are equal.]

When AD = BC, AO = OB, mBAO = 45, mBAC = 90, ABDC becomes a square.

BC need not be the perpendicular bisector of AD always. BC will be perpendicular to AD always. BC will be the perpendicular bisector only if AB = BD.

AD will be the perpendicular bisector of BC always as AB = AC.

Only statements 1, 2 and 4 are correct.

6.
What shall be the value of $x$ so that AE will be parallel to BC, if $y$ = 106?

 a. 72 b. 76 c. 74 d. 78

#### Solution:

mDBC = 106
[Given.]

mDBC = mBAE
[Corresponding Angles.]

mABC = mACB = 74
[DBC & ABC are Supplementary angles, AB = AC, mABC = mACB.]

mBAC = 180 - 2 × 74 = 32

x = 106 - 32 = 74
[Step 2.]

7.
In ΔABC, BD : DC is

 a. 1 : 1 b. 2 : 1 c. 1 : 2 d. 3 : 1

#### Solution:

In an Isosceles triangle, the perpendicular drawn from the vertex onto the base, bisects the base.

[Step 1.]

BD : DC = 1 : 1

8.
In the figure, if $\angle$BAD = 31o, then $\angle$DAC equals

 a. 33o b. 31o c. 29o d. 59o

#### Solution:

In an Isosceles triangle, the perpendicular drawn from the vertex to the base is the bisector of the vertex angle.

[Step 1.]

DAC = 31o

9.
In Δ$A$$B$$C$, if $x$ = 64, then find $m$$\angle$$B$.

 a. 58 b. 68 c. 63 d. 53

#### Solution:

AB = AC B = C
[Isosceles triangle theorem.]

mA + mB + mC = 180
[Triangle-Angle-Sum theorem.]

64 + mB + mB = 180
[C = B.]

2mB = 116
[Simplify.]

mB = 58
[Solve for B.]

10.
In ΔABC, $m$$\angle$A = $m$$\angle$B. Find the perimeter of ΔABC if $x$ = 8 cm and $y$ = 10 cm.

 a. 26 cm b. 32 cm c. 28 cm d. 34 cm

#### Solution:

In ΔABC, mA = mB AC = BC
[Converse of Isosceles triangle theorem.]

Perimeter of ΔABC = AB + BC + CA
[Definition.]

Perimeter of ΔABC = 8 + 10 + 10 = 28 cm