triangle ABC is isosceles with AC = BC and <ACB = 106 degrees. Point M is in the interior of the triangle so that <MAC = 7 degrees and <MCA = 23 degrees. Find the number of degrees in <CMB
90
V, this was a waste of time answering it.
To find the measure of angle CMB, we need to use the properties of angles in a triangle. Here's how we can do it step by step:
Step 1: Draw a diagram
It's always helpful to have a visual representation of the problem. Draw a triangle ABC, with AC = BC, and angle ACB measuring 106 degrees. Place point M inside the triangle such that angle MAC measures 7 degrees and angle MCA measures 23 degrees.
Step 2: Identify what we know
From the problem statement, we know that triangle ABC is isosceles, meaning AC = BC. We are also given the measures of angles MAC and MCA.
Step 3: Use the properties of an isosceles triangle
Since triangle ABC is isosceles with AC = BC, we know that angles MAC and MCA are congruent. Therefore, angle MCA also measures 7 degrees.
Step 4: Use the Triangle Angle Sum Theorem
The Triangle Angle Sum Theorem states that the sum of angles in a triangle is always 180 degrees. Since we know angles MAC, MCA, and ACB, we can find angle BAC by subtracting the sum of angles MAC and MCA from 180 degrees.
Angle BAC = 180 degrees - (angle MAC + angle MCA)
= 180 degrees - (7 degrees + 7 degrees)
= 180 degrees - 14 degrees
= 166 degrees
Step 5: Use the properties of a straight line
Angles along a straight line add up to 180 degrees. In this case, angle ACB lies in a straight line with angle BAC.
Angle BAC + angle ACB = 180 degrees
166 degrees + angle ACB = 180 degrees
angle ACB = 180 degrees - 166 degrees
angle ACB = 14 degrees
Step 6: Use the Triangle Angle Sum Theorem again
Now that we know all three angles in triangle CMB (angle MCB, angle CMB, and angle CMB), we can find the measure of angle CMB by subtracting the sum of angles MCB and MBC from 180 degrees.
angle CMB = 180 degrees - (angle MCB + angle MBC)
= 180 degrees - (23 degrees + 14 degrees)
= 180 degrees - 37 degrees
= 143 degrees
So, the measure of angle CMB is 143 degrees.