What is the solution with steps---**Question Number:** 4.
**Question Stem:**
The measure of angle ABC is 90°. What is the measure of angle ABC?
**Options:**
A) 40 degrees
B) 45 degrees
C) 30 degrees
D) 55 degrees
**Chart/Diagram Description:**
* **Type:** Geometric figure - a right triangle.
* **Main Elements:**
* **Vertices:** Labeled A, B, and C.
* **Angles:**
* Angle at vertex B is marked with a square, indicating a right angle (90 degrees).
* Angle at vertex A (∠BAC) is labeled as 6x.
* Angle at vertex C (∠BCA) is labeled as 3x.
* **Sides:** The triangle has sides connecting the vertices A, B, and C. The side AB appears vertical and BC appears horizontal, forming the right angle at B.
**Other Relevant Text:**
* Handwritten calculation below the diagram: 90 ÷ 3 = 30 ?
* Option C (30 degrees) is circled.
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We have a right triangle ABC where angle B is 90 degrees. Angle A is labeled as 6x and angle C is labeled as 3x. We need to find the measure of angle C. Let's start by using the fact that the sum of angles in any triangle equals 180 degrees.
Now let's set up our equation. First, we know that the sum of all angles in a triangle equals 180 degrees. So angle A plus angle B plus angle C equals 180 degrees. Next, we substitute our given values: 6x plus 90 degrees plus 3x equals 180 degrees. Finally, we combine like terms: 6x plus 3x gives us 9x, so we have 9x plus 90 degrees equals 180 degrees.
Now let's solve for x. Step 4: We subtract 90 degrees from both sides of the equation. Starting with 9x plus 90 degrees equals 180 degrees, we get 9x equals 180 degrees minus 90 degrees, which simplifies to 9x equals 90 degrees. Step 5: We divide both sides by 9. So x equals 90 degrees divided by 9, which gives us x equals 10 degrees.
Now we can find the measure of angle C. Step 6: We calculate angle C by substituting our value of x. Since angle C equals 3x, and x equals 10 degrees, we have angle C equals 3 times 10 degrees, which equals 30 degrees. Therefore, the measure of angle C is 30 degrees. Looking at our multiple choice options, the answer is C, 30 degrees.
Let's summarize our complete solution. We started with the fact that the sum of angles in any triangle equals 180 degrees. We substituted our given values to get 6x plus 90 degrees plus 3x equals 180 degrees. We combined like terms to get 9x plus 90 degrees equals 180 degrees. We solved for x by subtracting 90 degrees from both sides, giving us 9x equals 90 degrees. Dividing by 9, we found that x equals 10 degrees. Finally, we calculated angle C as 3 times x, which equals 30 degrees. We can verify our answer: 60 degrees plus 90 degrees plus 30 degrees equals 180 degrees. Therefore, the correct answer is C, 30 degrees.