Question

a. If $\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{35}$, find$\mathbf{m}\angle \mathbf{ABC}$

b. If$\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{k}$ find $\mathbf{m}\angle \mathbf{ABC}$.

Short answer

1. The measures of$\angle ABC$ is $\mathbf{90}$.
2. The measures of$\angle ABC$ is $\mathbf{90}$.

Step-by-step solution

Part a. 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 $\mathrm{\Delta }ABD$, in which$AD\cong BD$ then by isosceles triangle theorem, $\angle 1\cong \angle 2$, that is

$m\angle 1=m\angle 2=35$.

Part a. 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,

$$\begin{gathered} m\angle 3=m\angle 1+m\angle 2 \\ =35+35 \\ =70 \end{gathered}$$

Part a. Step 3. Apply isosceles triangle theorem.

Consider $\mathrm{\Delta }DBC$, in which$DB\cong DC$ then by isosceles triangle theorem, $\angle 4\cong \angle 5$, that is

$m\angle 4=m\angle 5$.

Part a. Step 4. Apply an angle sum theorem.

The sum of measures of all the angles of a triangle is 180.

Consider $\mathrm{\Delta }DBC$, such that,

$$\begin{gathered} m\angle 5+m\angle 4+m\angle 3=180 \\ m\angle 5+m\angle 5+70=180 \\ 2m\angle 5=110 \\ m\angle 5=55 \end{gathered}$$

Part a. Step 5. Description of step.

Substitute 35 for $m\angle 2$and 55 for $m\angle 5$.

$$\begin{gathered} m\angle ABC=m\angle 2+m\angle 5 \\ =35+55 \\ =90 \end{gathered}$$

Therefore, the measure of$\angle ABC$ is 90.

Part b. Step 1. Apply isosceles triangle theorem.

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

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

Part b. Step 2. Apply exterior angle theorem.

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 then by exterior angle theorem,

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

Part b. Step 3. Apply isosceles triangle theorem.

Consider $\mathrm{\Delta }DBC$, in which$DB\cong DC$ then by isosceles triangle theorem, $\angle 4\cong \angle 5$, that is

$m\angle 4=m\angle 5$.

Part b. Step 4. Apply an angle sum theorem.

Consider $\mathrm{\Delta }DBC$then by angle sum theorem,

$$\begin{gathered} m\angle 5+m\angle 4+m\angle 3=180 \\ m\angle 5+m\angle 5+2k=180 \\ 2m\angle 5+2k=180 \\ m\angle 5+k=90 \\ m\angle 5=90-k \end{gathered}$$

Part b. Step 5. Description of step.

Substitute $k$for $m\angle 2$and $90-k$for $m\angle 5$.

$$\begin{gathered} m\angle ABC=m\angle 2+m\angle 5 \\ =k+90-k \\ =90 \end{gathered}$$

Therefore, the measure of$\angle ABC$ is 90.