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. Which of the following completes the proof? Alternate Interior Angles Theorem Converse of the Alternate Interior Angles Theorem Converse of the Same-Side Interior Angles Theorem Same-Side Interior Angles Theorem
2 answers:
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.
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Answer:
Rs 93.75
Step-by-step explanation:
Solution:
A man starts from his house with some money to visit four temples.
<u>1. As soon as he enters temple 1 his money gets doubled. </u>
Let he had money at first =
After enter the 1st temple his money =
He donated = Rs 100
Money left after donation at 1st temple =
<u>2. At temple 2 again his money is doubled and he donates Rs 100</u>
He had at 2nd temple =
Donated = Rs 100
Money left after donation at 2nd temple =
<u>3. His money is doubled again at temple 3 and he donates Rs 100.</u>
He had at 3rd temple =
Donated = Rs 100
Money left after donation at 3rd temple =
<u>4. At temple 4 again his money gets doubled. He donates 100 rs. again and returns home empty handed.</u>
He had at 4th temple =
Donated = Rs 100
Money left after donation at 4th temple =
<u>As he returned empty hand or with zero money, equation will be:-</u>
Thus, he had Rs 93.75 when he started from his house.
Answer:
The length of legs lm, mn, and lk is 32 while that of kn is 36.
Step-by-step explanation:
The diagram is attached in the answer.
As both the angles are equal thus the two forms isosceles triangles. This is true because the converse of base angle theorem is applicable here. The theorem clearly states that when a trapezoid is divided by the diagonal, it forms two set of angles where m<1=m<2 and m<lmk=m<nmk
This leads the two set of legs such as
Or this could be written as
Now the remaining two sides are given as
as this form an isosceles triangle so
Similarly
Or
Now the perimeter is given as
Here perimeter is given as 132 so this gives:
So the four legs are of length as given below:
So the length of legs lm, mn, and lk is 32 while that of kn is 36.
