Question

angle restrictions based on side lengths

isosceles triangles can be acute, ____, or obtuse.

consider the triangles in the figure.

• δstq: this is an ____ triangle because all the angles are less than 90°. since tq ≅ qs, it’s an isosceles triangle. so, it’s an isosceles acute triangle.

• δpqr: this is a right isosceles triangle.

• δsqp: angle q is an obtuse angle. since sq ≅ qp, it’s an obtuse isosceles triangle.

Explanation:

First Blank (Isosceles triangles can be acute, ____, or obtuse)

Step1: Recall triangle angle classifications

Triangles are classified by angles as acute (all angles < 90°), right (one angle = 90°), or obtuse (one angle > 90°). Isosceles triangles (with at least two equal sides) can fall into any of these angle - based categories. Given the options of acute, [blank], obtuse, and knowing the three angle - based triangle types, the missing one is "right".

Step2: Verify with the example

The example later mentions a right isosceles triangle ($\triangle PQR$), which confirms that right is a valid angle - based classification for isosceles triangles.

Second Blank ($\triangle STQ$: This is an ____ triangle because all the angles are less than $90^{\circ}$)

Step1: Recall the definition of an acute triangle

A triangle with all angles less than $90^{\circ}$ is defined as an acute triangle. Since the problem states that for $\triangle STQ$ all angles are less than $90^{\circ}$, by the definition of an acute triangle, this triangle is acute.

Answer:

s:
First blank: right
Second blank: acute