•line a || line b

•line c || line d

•segment eg || line e

We have to prove that rectangles are parallelograms with congruent Diagonals.

Solution:

1. ∠R=∠E=∠C=∠T=90°

2. ER= CT, EC ║RT

3. Diagonals E T and C R are drawn.

4. Shows Quadrilateral R E CT is a Rectangle.→→[Because if in a Quadrilateral One pair of Opposite sides are equal and parallel and each of the interior angle is right angle than it is a Rectangle.]

5. Quadrilateral RECT is a Parallelogram.→→[If in a Quadrilateral one pair of opposite sides are equal and parallel then it is a Parallelogram]

6. In Δ ERT and Δ CTR

(a) ER= CT→→[Opposite sides of parallelogram]

(b) ∠R + ∠T= 90° + 90°=180°→→→Because RECT is a rectangle, so ∠R=∠T=90°]

(c) Side TR is Common.

So, Δ ERT ≅ Δ CTR→→[SAS]

Diagonal ET= Diagonal CR →→→[CPCTC]

In step 6, while proving Δ E RT ≅ Δ CTR, we have used

(b) ∠R + ∠T= 90° + 90°=180°→→→Because RECT is a rectangle, so ∠R=∠T=90°]

Here we have used ,Option (D) : Same-Side Interior Angles Theorem, which states that Sum of interior angles on same side of Transversal is supplementary.

The answer is segment EC is parallel to segment RT.

Spencer wrote the following paragraph proof showing that rectangles are parallelograms with congruent diagonals.

Quadrilateral R E C T is shown with right angles at each of the four corners. Segments E R and C T have single hash marks indicating they are congruent while segments E C and R T have two arrows indicating they are parallel. Segments E T and C R are drawn.

According to the given information, quadrilateral RECT is a rectangle. By the definition of a rectangle, all four angles measure 90°. Segment ER is parallel to segment CT and segment EC is parallel to segment RT by the . Quadrilateral RECT is then a parallelogram by definition of a parallelogram. Now, construct diagonals ET and CR. Because RECT is a parallelogram, opposite sides are congruent. Therefore, one can say that segment ER is congruent to segment CT. Segment TR is congruent to itself by the Reflexive Property of Equality. The Side-Angle-Side (SAS) Theorem says triangle ERT is congruent to triangle CTR. And because corresponding parts of congruent triangles are congruent (CPCTC), diagonals ET and CR are congruent.

hope this helps

I'll start with the triangle figure.

Interior angles of a triangle is equal to 180°. The given triangle is an Isosceles triangle. It has 2 equal sides and 2 equal angles.

#7 is 69° because its vertical angle is 69°. Vertical angles are equal.

#8 is 111°. As I said, isosceles triangle has 2 equal angles. The angle that is beside #8 is 69°, equal to #7. So, 180° - 69° = 111°

Interior angles are already given except for #4. So, 180° - 69° - 69° = 42°

#3 is computed by 180° - 42° - 69° = 69°

#9 = 116° ; alternate interior angles are equal.

#10 = 180° - 116° = 64°

The last triangle looks like an equilateral triangle. It means that its sides and angles are equal.

360° / 6 = 60°

#1, #2, #5, and #6 = 60° each

#11 = 60° - equilateral triangle, all interior angles are equal.

#12 = 30°; 180° - 90° - 60° = 30°

Interior angles of a triangle is equal to 180°. The given triangle is an Isosceles triangle. It has 2 equal sides and 2 equal angles.

#7 is 69° because its vertical angle is 69°. Vertical angles are equal.

#8 is 111°. As I said, isosceles triangle has 2 equal angles. The angle that is beside #8 is 69°, equal to #7. So, 180° - 69° = 111°

Interior angles are already given except for #4. So, 180° - 69° - 69° = 42°

#3 is computed by 180° - 42° - 69° = 69°

#9 = 116° ; alternate interior angles are equal.

#10 = 180° - 116° = 64°

The last triangle looks like an equilateral triangle. It means that its sides and angles are equal.

360° / 6 = 60°

#1, #2, #5, and #6 = 60° each

#11 = 60° - equilateral triangle, all interior angles are equal.

#12 = 30°; 180° - 90° - 60° = 30°

Interior angles of a triangle is equal to 180°. The given triangle is an Isosceles triangle. It has 2 equal sides and 2 equal angles.

#7 is 69° because its vertical angle is 69°. Vertical angles are equal.

#8 is 111°. As I said, isosceles triangle has 2 equal angles. The angle that is beside #8 is 69°, equal to #7. So, 180° - 69° = 111°

Interior angles are already given except for #4. So, 180° - 69° - 69° = 42°

#3 is computed by 180° - 42° - 69° = 69°

#9 = 116° ; alternate interior angles are equal.

#10 = 180° - 116° = 64°

The last triangle looks like an equilateral triangle. It means that its sides and angles are equal.

360° / 6 = 60°

#1, #2, #5, and #6 = 60° each

#11 = 60° - equilateral triangle, all interior angles are equal.

#12 = 30°; 180° - 90° - 60° = 30°

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