Question

1. a triangle has angle measures of x + 15, 3x - 35, and 4x.

a. find the value of x.
b. find the measure of each angle.
c. what type of triangle is it? why?

1. what is the difference between an isosceles triangle and an isosceles right triangle? (draw an example of both)

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So, $(x + 15)+(3x - 35)+4x=180$.
Combining like terms, we get $x+3x + 4x+15 - 35=180$, which simplifies to $8x-20 = 180$.

Step2: Solve for $x$

Add 20 to both sides of the equation: $8x-20 + 20=180 + 20$, so $8x=200$.
Divide both sides by 8: $x=\frac{200}{8}=22.5$.

Step3: Find the measure of each angle

For the first angle: $x + 15$, substituting $x = 22.5$, we get $22.5+15=37.5^{\circ}$.
For the second angle: $3x - 35$, substituting $x = 22.5$, we get $3\times22.5-35=67.5 - 35=32.5^{\circ}$.
For the third angle: $4x$, substituting $x = 22.5$, we get $4\times22.5 = 90^{\circ}$.

Step4: Classify the triangle

Since one of the angles is $90^{\circ}$, the triangle is a right - triangle.

Answer:

a. $x = 22.5$
b. The angles are $37.5^{\circ}$, $32.5^{\circ}$, and $90^{\circ}$
c. It is a right - triangle because one of its angles measures $90^{\circ}$