Question

3 multiple choice 4 points
the measure of one acute angle of an obtuse triangle is ( 20^circ ) more than the measure of the other acute angle. what is a possible measure of the smaller acute angle?
( 35^circ )
( 40^circ )
( 30^circ )
( 45^circ )

4 multiple choice 4 points
in this triangle, what is the measure of ( angle c )?
image of a right triangle with right angle at b, angle at a is ( 58^circ ), angle at c is unknown
( 122^circ )
( 58^circ )
( 42^circ )
( 32^circ )

Explanation:

Step1: Define variables for Q3

Let $x$ = smaller acute angle, $x+20^\circ$ = larger acute angle.

Step2: Set up inequality for Q3

Sum of acute angles $< 90^\circ$ (obtuse triangle):
$x + (x+20^\circ) < 90^\circ$

Step3: Solve inequality for Q3

$2x + 20^\circ < 90^\circ$
$2x < 70^\circ$
$x < 35^\circ$
Only 30° fits.

Step4: Use triangle sum for Q4

Right triangle: $\angle A + \angle C = 90^\circ$

Step5: Calculate $\angle C$ for Q4

$\angle C = 90^\circ - 58^\circ = 32^\circ$

Answer:

1. 30°
2. 32°