Question

Complete.

a. If$\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{20}$ then$\mathbf{m}\angle \mathbf{3}\mathbf{=}\mathbf{?},\mathbf{m}\angle \mathbf{4}\mathbf{=}\mathbf{?}$ and$\mathbf{m}\angle \mathbf{5}\mathbf{=}\mathbf{?}$

b. If$\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{x}$ then$\mathbf{m}\angle \mathbf{3}\mathbf{=}\mathbf{?},\mathbf{m}\angle \mathbf{4}\mathbf{=}\mathbf{?}$ and$\mathbf{m}\angle \mathbf{5}\mathbf{=}\mathbf{?}$

Short answer

1. The measures of$\angle 3,\angle 4\text{and}\angle \text{5}$ are$\mathbf{40}\mathbf{,}\mathbf{40}\text{and}\mathbf{60}$ respectively.
2. The measures of$\angle 3,\angle 4\text{and}\angle \text{5}$ are$\mathbf{2}\mathbf{x}\mathbf{,}\mathbf{2}\mathbf{x}\text{and}\mathbf{3}\mathbf{x}$ respectively.

Step-by-step solution

Part a. Step 1. Label the diagram.

Part a. Step 2. Apply isosceles triangle theorem.

If the two sides of a triangle are congruent then the angles opposite to those sides are congruent.

Consider $\mathrm{\Delta }BDC$, in which$BD\cong DC$ then by isosceles triangle theorem, $\angle 1\cong \angle 2$, that is

$m\angle 1=m\angle 2=20$.

Part a. Step 3. Apply exterior angle theorem.

The measure of exterior angle is equal to the sum of measure of two remote interior angles of a triangle.

From the given figure, it can be observed that is$\angle 3$ exterior angle and$\angle 1\text{and}\angle \text{2}$ are remote exterior angles, such that,

$$\begin{gathered} m\angle 3=m\angle 1+m\angle 2 \\ =20+20 \\ =40 \end{gathered}$$

Part a. Step 4. Apply isosceles triangle theorem.

Consider $\mathrm{\Delta }ABD$, in which$AB\cong BD$ then by isosceles triangle theorem, $\angle 4\cong \angle 3$, that is,

$m\angle 4=m\angle 3=40$.

Part a. Step 5. Apply exterior angle theorem.

From the given figure, it can be observed that$\angle 5$ is exterior angle and$\angle 4\text{and}\angle 1$ are remote exterior angles, such that,

$$\begin{gathered} m\angle 5=m\angle 1+m\angle 4 \\ =20+40 \\ =60 \end{gathered}$$

Therefore, the measures of$\angle 3,\angle 4\text{and}\angle \text{5}$ are$40,40\text{and60}$ respectively.

Part b. Step 1. Apply isosceles triangle theorem.

If the two sides of a triangle are congruent then the angles opposite to those sides are congruent.

Consider $\mathrm{\Delta }BDC$, in which$BD\cong DC$ then by isosceles triangle theorem, $\angle 1\cong \angle 2$, that is

$m\angle 1=m\angle 2=x$.

Part b. Step 2. Apply exterior angle theorem.

The measure of exterior angle is equal to the sum of measure of two remote interior angles of a triangle.

From the given figure, it can be observed that $\angle 3$is exterior angle and $\angle 1\text{and}\angle \text{2}$are remote exterior angles, such that,

$$\begin{gathered} m\angle 3=m\angle 1+m\angle 2 \\ =x+x \\ =2x \end{gathered}$$

Part b. Step 3. Apply isosceles triangle theorem.

Consider $\mathrm{\Delta }ABD$, in which$AB\cong BD$ then by isosceles triangle theorem, $\angle 4\cong \angle 3$, that is,

$m\angle 4=m\angle 3=2x$.

Part b. Step 4. Apply exterior angle theorem.

From the given figure, it can be observed that $\angle 5$is exterior angle and $\angle 4\text{and}\angle 1$are remote exterior angles, such that,

$$\begin{gathered} m\angle 5=m\angle 1+m\angle 4 \\ =x+2x \\ =3x \end{gathered}$$

Therefore, the measures of$\angle 3,\angle 4\text{and}\angle \text{5}$ are$2x,2x\text{and3x}$ respectively.