Question

(4 - 32) use what you know about the sum of the angles of a triangle to find m < abc and m < bac.

• are these angles acute or obtuse?
• find the sum of these two angles and the name of their relationship.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. In right - triangle $ABC$ with $\angle C = 90^{\circ}$, we have $(6x + 8^{\circ})+(2x+2^{\circ})+90^{\circ}=180^{\circ}$.

Step2: Simplify the equation

Combine like terms: $6x+2x + 8^{\circ}+2^{\circ}+90^{\circ}=180^{\circ}$, which gives $8x+100^{\circ}=180^{\circ}$.

Step3: Solve for $x$

Subtract $100^{\circ}$ from both sides: $8x=180^{\circ}- 100^{\circ}=80^{\circ}$. Then divide by 8: $x = 10^{\circ}$.

Step4: Find $\angle ABC$ and $\angle BAC$

For $\angle ABC=2x + 2^{\circ}$, substitute $x = 10^{\circ}$: $\angle ABC=2\times10^{\circ}+2^{\circ}=22^{\circ}$.
For $\angle BAC=6x + 8^{\circ}$, substitute $x = 10^{\circ}$: $\angle BAC=6\times10^{\circ}+8^{\circ}=68^{\circ}$.

Step5: Determine if angles are acute or obtuse

Since $0^{\circ}<22^{\circ}<90^{\circ}$ and $0^{\circ}<68^{\circ}<90^{\circ}$, both $\angle ABC$ and $\angle BAC$ are acute.

Step6: Find the sum of the two angles and their relationship

$\angle ABC+\angle BAC=22^{\circ}+68^{\circ}=90^{\circ}$. They are complementary angles.

Answer:

$m\angle ABC = 22^{\circ}$, $m\angle BAC=68^{\circ}$; both angles are acute; the sum of the two angles is $90^{\circ}$ and they are complementary.