Question

b. determine the measure of each unknown angle. list the side lengths from shortest to longest for each diagram. explain your reasoning.
1.
2.
3.
4.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°.

Step2: Find unknown angles

For the first triangle with angles 47° and 22°

Let the unknown angle be $\theta_1$. Then $\theta_1=180-(47 + 22)=111^{\circ}$.

For the second triangle with angles 55° and 59°

Let the unknown angle be $\theta_2$. Then $\theta_2=180-(55 + 59)=66^{\circ}$.

For the third triangle with angles 26° and 120°

Let the unknown angle be $\theta_3$. Then $\theta_3=180-(26 + 120)=34^{\circ}$.

For the fourth triangle with angles 40° and 86°

Let the unknown angle be $\theta_4$. Then $\theta_4=180-(40 + 86)=54^{\circ}$.

Step3: Recall side - angle relationship in a triangle

In a triangle, the side opposite the smallest angle is the shortest and the side opposite the largest angle is the longest.

For the first triangle

The angles are 22°, 47°, 111°. So the sides from shortest to longest are: the side opposite 22° (say $c$), the side opposite 47° (say $a$), the side opposite 111° (say $b$), i.e., $c,a,b$.

For the second triangle

The angles are 55°, 59°, 66°. So the sides from shortest to longest are: the side opposite 55° (say $r$), the side opposite 59° (say $t$), the side opposite 66° (say $s$), i.e., $r,t,s$.

For the third triangle

The angles are 26°, 34°, 120°. So the sides from shortest to longest are: the side opposite 26° (say $l$), the side opposite 34° (say $k$), the side opposite 120° (say $m$), i.e., $l,k,m$.

For the fourth triangle

The angles are 40°, 54°, 86°. So the sides from shortest to longest are: the side opposite 40° (say $y$), the side opposite 54° (say $x$), the side opposite 86° (say $z$), i.e., $y,x,z$.

Answer:

1. Unknown angle: 111°, Sides from shortest to longest: $c,a,b$
2. Unknown angle: 66°, Sides from shortest to longest: $r,t,s$
3. Unknown angle: 34°, Sides from shortest to longest: $l,k,m$
4. Unknown angle: 54°, Sides from shortest to longest: $y,x,z$