Answer:
To show that E₂ = 180 - 3x, we need to use the properties of a parallelogram and the given information.
1) Since PQRS is a parallelogram, opposite angles are equal. Therefore, PQE = SRE = 3x.
2) We are given that PQ = PE, QE = QR, ER = SR. Since opposite sides of a parallelogram are equal, we can say that PQ = SR and PE = ER.
Using the given information, we can construct the following diagram:
P_____Q
|\ |\
| \ | \
| \ | \
|___\_|___\
E R S
Now, let's examine angle EQS. Since PQ = SR and QE = ER, triangle EQS is an isosceles triangle.
In an isosceles triangle, the base angles are equal. Therefore, angle EQS = angle ERQ.
We know that angle PQE = 3x, so angle ERQ = 3x as well.
Now, let's examine angle E₂. Since PQ = PE and QR = RE, triangle PQR is also an isosceles triangle.
In an isosceles triangle, the base angles are equal. Therefore, angle PQR = angle PRQ.
We know that angle PQE = 3x, so angle PRQ = 3x as well.
Since angle PQR = angle PRQ, we can conclude that angle E₂ = angle ERQ + angle PRQ = 3x + 3x = 6x.
To find E₂, we can subtract angle E₂ from 180 degrees since the sum of angles in a triangle is 180 degrees.
Therefore, E₂ = 180 - 6x = 180 - 3x - 3x = 180 - 3x.
So, we have shown that E₂ = 180 - 3x.
Now, let's move on to calculating the size of QÊR.
Since PQ = QR and PQE = 3x, we can say that triangle PQE is an isosceles triangle.
In an isosceles triangle, the base angles are equal. Therefore, angle QPE = angle PQE = 3x.
Since opposite angles of a parallelogram are equal, we can conclude that angle QÊR = angle QPE = 3x.
So, the size of QÊR is 3x.
Explanation:
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