Question

an exterior angle at the base of an isosceles triangle is always
select one:
a. a right angle
b. an obtuse angle
c. an acute angle

if the angles of a triangle are represented by x, y and (x + y), the triangle is
select one:
a. none of these
b. an isosceles triangle
c. an equilateral triangle
d. a right triangle

Explanation:

Step1: Analyze first triangle property

Let the base angles of the isosceles triangle be $\alpha$, so $\alpha < 90^\circ$ (since sum of triangle angles is $180^\circ$, two base angles can't be $\geq90^\circ$). The exterior angle at the base is $180^\circ - \alpha$. Since $\alpha < 90^\circ$, $180^\circ - \alpha > 90^\circ$, so it is obtuse.

Step2: Analyze second triangle angles

Sum of triangle angles is $180^\circ$. So $x + y + (x+y) = 180^\circ$. Simplify: $2(x+y)=180^\circ$, so $x+y=90^\circ$. One angle is $90^\circ$, so it is a right triangle.

Answer:

1. b. an obtuse angle
2. d. a right triangle