Question

Can someone help me figure out the 2 proofs for questions 1 and 3 and help me find what X and Y equal for question 2 if you can!! In geometry

Answer 1

Hello! We can prove this with some statements, look:

• Notice that CD is parallel to AB;

,

• angle 1 ≅ angle 2

,

• M is the midpoint of AB;

Statement 1:

`\hat{\text{AMC}}\cong\hat{\text{BMD}}`

Reasoning 1:

As M is the midpoint of AB and we have two similar lines starting in M, we can divide the angle M into two equal angles m1 and m2.

definition of midpoint

Statement 2:

`\hat{C}=\hat{D}`

Reasoning 2:

As we already know that angle 1 ≅ angle 2, let's calculate the sum of the angles C and D, look:

angle C:

90º + angle 1

angle D:

90º + angle 2

so, 90º + angle 1 ≅ 90º + angle 2, it means that angle C ≅ angle D.

Statement 3:

`\hat{MAC}\text{ = }\hat{\text{MBD}}`

Reasoning 3:

The sum of the internal angles of a triangle must be equal to 180º, right? So, knowing it we can say that angles A and B are equal, look:

m1 + A + C = m2 + B + D = 180º

remember, m1 ≅ m2 and C ≅ D, so A ≅ B too.

According to the explanation and image, we can prove that triangle CAM ≅ triangle DBM.

Table: