Question

Can anyone explain this
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Answer 1

To find the size of ∠LAMB, we need to use the properties of angles in a triangle and the given information.

In triangle ABC, we are told that M is the midpoint of BC. This means that AM is equal in length to MC.

We are also given that AT bisects ∠BAC, which means that angle BAT is equal to angle CAT.

Since we know that ATC = 56°, and angle BAT is equal to angle CAT, we can determine that each of these angles is half of 56°, which is 28°.

Now, let's consider triangle ABM. We know that angles in a triangle add up to 180°. We already have angle BAM, which is 28°. Since M is the midpoint of BC, angle ABM is equal to angle CBM. Therefore, angle ABM + angle BAM + angle CBM = 180°.

Substituting the known values, we have:

28° + 28° + angle CBM = 180°.

Simplifying the equation, we find that:

56° + angle CBM = 180°.

To isolate angle CBM, we subtract 56° from both sides:

angle CBM = 180° - 56° = 124°.

Therefore, the size of ∠LAMB is 124°.

So, the correct answer is (E) None of the previous alternatives.

Explanation:

Answer 2

The measure of angle AMB is 62 degrees, i.e. option E is the correct option.

In triangle ABC, M is the midpoint of BC, AM = MC, and AT bisects angle BAC.

Angle ATC is 56 degrees.

We need to find the measure of angle AMB.

Steps to solve:

Triangle AMT is isosceles: Since AM = MC and AT is the angle bisector, triangle AMT is isosceles. This means that angles ATM and AMT are congruent.

Angles in triangle AMT: We know that the angles in a triangle add up to 180 degrees. Therefore, angle ATM + angle AMT + angle MAT = 180 degrees. Since angles ATM and AMT are congruent, we can substitute 2 * angle ATM for them in the equation. This gives us: 2 * angle ATM + angle MAT = 180 degrees.

Angle MAT: We are given that angle ATC is 56 degrees. Angle MAT and angle ATC are vertical angles, which means they are congruent. Therefore, angle MAT = 56 degrees.

Angle ATM: Substitute angle MAT = 56 degrees into the equation from step 2: 2 * angle ATM + 56 degrees = 180 degrees. Solve for angle ATM: 2 * angle ATM = 124 degrees. Therefore, angle ATM = 62 degrees.

Angle AMB: Angle AMB and angle ATM are alternate interior angles formed by transversal TM and lines AB and AC. Therefore, they are congruent. So, angle AMB = angle ATM = 62 degrees.