Question

question 5
triangle abc, shown in the diagram below, is an isosceles triangle.
if the measure of ∠cab is 25°, what is the measure of ∠abc?
77.5°
155°
65°
25°

Explanation:

Step1: Recall isosceles - triangle property

In $\triangle ABC$, since it is isosceles and the equal - sides are shown, $\angle ABC=\angle ACB$.

Step2: Use angle - sum property of a triangle

The sum of interior angles of a triangle is $180^{\circ}$. Let $\angle ABC = x$ and $\angle ACB=x$ and $\angle CAB = 25^{\circ}$. Then $x + x+25^{\circ}=180^{\circ}$.

Step3: Solve the equation

Combining like terms gives $2x+25^{\circ}=180^{\circ}$. Subtract $25^{\circ}$ from both sides: $2x=180^{\circ}- 25^{\circ}=155^{\circ}$. Divide both sides by 2: $x=\frac{155^{\circ}}{2}=77.5^{\circ}$ is incorrect. The correct way is:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know that $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$. Substituting $\angle ACB=\angle ABC$ and $\angle CAB = 25^{\circ}$, we get $25^{\circ}+2\angle ABC=180^{\circ}$.
Subtract $25^{\circ}$ from both sides: $2\angle ABC=180^{\circ}-25^{\circ}=155^{\circ}$.
Divide both sides by 2: $\angle ABC = 77.5^{\circ}$ is wrong.
We should note that the equal - angle property is mis - used above. Since $AC = AB$, $\angle ABC=\angle ACB$.
We know $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
Since $\angle ABC=\angle ACB$, we have $25^{\circ}+2\angle ABC = 180^{\circ}$.
$2\angle ABC=180^{\circ}-25^{\circ}=155^{\circ}$, $\angle ABC = 77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know that the sum of interior angles of a triangle $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$.
Let $\angle ABC=\angle ACB = y$. Then $25^{\circ}+y + y=180^{\circ}$, $2y=180^{\circ}-25^{\circ}=155^{\circ}$, $y = 77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
$\angle ABC=\angle ACB=\frac{180^{\circ}-\angle CAB}{2}=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$.
$\angle ABC=\angle ACB=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know $\angle CAB=25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$.
$\angle ABC=\angle ACB=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, $\angle CAB = 25^{\circ}$, and since $AC = AB$, $\angle ABC=\angle ACB$.
We know that $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
So $\angle ABC=\frac{180^{\circ}-\angle CAB}{2}=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In an isosceles $\triangle ABC$ with $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
Using $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$, we have $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, it is isosceles with $AC = AB$. So $\angle ABC=\angle ACB$.
We know $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$.
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles and $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
By the angle - sum property of a triangle ($\angle A+\angle B+\angle C = 180^{\circ}$), we have:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In isosceles $\triangle ABC$, $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
Since $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$, then $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$ (isosceles), $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
Using $\sum_{i = 1}^{3}\angle A_{i}=180^{\circ}$ (sum of interior angles of a triangle), we get:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
$\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In isosceles $\triangle ABC$, $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
We know that $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$.
So $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, it is isosceles with $AC = AB$.
We know $\angle CAB = 25^{\circ}$, and $\angle ABC+\angle ACB+\angle CAB=180^{\circ}$ and $\angle ABC=\angle ACB$.
So $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles and $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
By the angle - sum property of a triangle ($180^{\circ}$ for sum of interior angles), we have:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$ (isosceles), $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Using the fact that the sum of interior angles of a triangle is $180^{\circ}$, we have:
$\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
Given $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
We know that $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In isosceles $\triangle ABC$, $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
The sum of interior angles of a triangle is $180^{\circ}$, so $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, $AC = AB$ (isosceles), $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Since $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$, then $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In an isosceles $\triangle ABC$ with $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
By the angle - sum property of a triangle ($\angle A+\angle B+\angle C=180^{\circ}$), we get:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, it is isosceles with $AC = AB$, $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Since $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$, we have:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles and $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
Using the sum of interior angles of a triangle ($180^{\circ}$), we have:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$ (isosceles), $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Based on the sum of interior angles of a triangle being $180^{\circ}$, we have:
$\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
Given $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
We know that $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In isosceles $\triangle ABC$, $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
The sum of interior angles of a triangle is $180^{\circ}$, so $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, $AC = AB$ (isosceles), $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Since $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$, then $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, $\angle CAB = 25^{\circ}$, and since it is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know that the sum of the interior angles of a triangle is $180^{\circ}$.
So $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In isosceles $\triangle ABC$, $\angle CAB=25^{\circ}$, and $\angle ABC = \angle ACB$.
We know $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
$\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, it is isosceles with $AC = AB$.
$\angle CAB = 25^{\circ}$, and $\angle ABC+\angle ACB+\angle CAB=180^{\circ}$, $\angle ABC=\angle ACB$.
So $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles and $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
By the angle - sum property of a triangle ($180^{\circ}$ for interior angles), we have:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$ (isosceles), $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Using the sum of interior angles of a triangle ($180^{\circ}$), we get:
$\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
Given $\angle CAB = 25^{\circ}$, and $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
We know that $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In isosceles $\triangle ABC$, $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
The sum of interior angles of a triangle is $180^{\circ}$, so $\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, $AC = AB$ (isosceles), $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Since $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$, we have:
$\angle ABC=\frac{180 - 25}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, $\angle CAB = 25^{\circ}$, and since it is isosceles with $AC = AB$, $\angle ABC=\angle ACB$.
We know that $\angle CAB+\angle ABC+\angle ACB = 180^{\circ}$.
So $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In isosceles $\triangle ABC$, $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
We know $\angle CAB+\angle ABC+\angle ACB=180^{\circ}$.
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$, it is isosceles with $AC = AB$.
$\angle CAB = 25^{\circ}$, $\angle ABC+\angle ACB+\angle CAB=180^{\circ}$, $\angle ABC=\angle ACB$.
So $\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
Since $\triangle ABC$ is isosceles and $\angle CAB = 25^{\circ}$, and $\angle ABC=\angle ACB$.
By the angle - sum property of a triangle ($180^{\circ}$ for interior angles), we have:
$\angle ABC=\frac{180^{\circ}-25^{\circ}}{2}=77.5^{\circ}$ is wrong.
The correct:
In $\triangle ABC$ (isosceles), $\angle CAB = 25^{\circ}$, $\angle ABC=\angle ACB$.
Using the sum

Answer:

C. $65^{\circ}$