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Problem 43

# (a)Let $$\mathrm{ABC}$$ be a right triangle with $\mathrm{m} \angle \mathrm{BCA}=90^{\circ}$ and $$\mathrm{m} \angle \mathrm{CAB}=30^{\circ}$$. What is $\mathrm{m} \angle \mathrm{ABC} ?$ (b) Prove that in a right triangle the sum of the measures of the angles adjacent to the hypotenuse is $$90^{\circ}$$.

Expert verified
$$m \angle ABC = 60^{\circ}$$ and the sum of the measures of the angles adjacent to the hypotenuse in a right triangle is always equal to $$90^{\circ}$$.
See the step by step solution

## Step 1: (a) Identifying given angles

In right triangle ABC, we're given that m∠BCA = 90° (making it the right angle) and m∠CAB = 30°. We need to find m∠ABC.

## Step 2: (a) Using the Triangle Sum Property

In any triangle, the sum of the interior angles is always equal to 180°. Therefore, m∠ABC + m∠BCA + m∠CAB = 180°.

## Step 3: (a) Finding the missing angle

Now we can plug in the values we're given and solve for m∠ABC: m∠ABC + 90° + 30° = 180° m∠ABC + 120° = 180° m∠ABC = 60° So, the measure of angle ABC is 60°.

## Step 4: (b) Identifying the hypotenuse

In right triangle ABC, the angle BCA has a measure of 90°, making the side AC the hypotenuse. We now have to prove that the sum of the measures of angles CAB and ABC is always 90°.

## Step 5: (b) Using the Triangle Sum Property

Just as in part (a), we will apply the Triangle Sum Property. In right triangle ABC, the sum of the interior angles m∠ABC + m∠BCA + m∠CAB = 180°.

## Step 6: (b) Proving the statement

Since we know that m∠BCA = 90°, we can substitute this into the equation: m∠ABC + 90° + m∠CAB = 180° Now, we can rearrange the equation to isolate angles ABC and CAB: (m∠ABC + m∠CAB) + 90° = 180° m∠ABC + m∠CAB = 90° Thus, we have proved that the sum of the measures of the angles adjacent to the hypotenuse in a right triangle is always equal to 90°.

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