Question

use the
your reasomng
sample problem
tommy drew a quadrilateral. he used a protractor to measure all
his angles of the quadrilateral.
how many pairs of angles must be congruent for the quadrilateral to
be a parallelogram?
opposite angles are co
pairs of opposite angles
. a parallelogram, so both
be congruent.
jol
the sum
kite with a perimeter of 28 ce
shorter sides is 10 centimeters
what are the lengths of each of the two longer sides?
→ enter the answer in the space provided. use numbers instead of
words.
first side
centimeters
second side
centimeters

Explanation:

First, let's solve the parallelogram angle problem:

Step1: Recall parallelogram angle property

In a parallelogram, opposite angles are congruent. A quadrilateral has 4 angles, forming 2 pairs of opposite angles.

Step2: Determine the number of congruent angle pairs

For a quadrilateral to be a parallelogram, both pairs of opposite angles must be congruent. So the number of pairs of congruent angles required is 2.

(for kite):

Step1: Recall kite side property

A kite has two pairs of adjacent equal sides. Let shorter sides sum to 10 (so each shorter is 5, but maybe the problem means each shorter side is 10? Wait, the text says "shorter sides is 10 centimeters" – maybe each shorter side is 10, but then perimeter would be \( 2*10 + 2l = 28 \) → \( 20 + 2l = 28 \) → \( 2l = 8 \) → \( l = 4 \), but that's shorter. So likely a misstatement, and the shorter sides sum to 10. So:

Step1: Let shorter side length sum to 10

Let the length of each shorter side be \( s \), so \( 2s = 10 \) → \( s = 5 \) cm.

Step2: Calculate longer side length

Perimeter \( P = 2s + 2l = 28 \). Substitute \( 2s = 10 \): \( 10 + 2l = 28 \).

Step3: Solve for \( l \)

\( 2l = 28 - 10 = 18 \) → \( l = \frac{18}{2} = 9 \) cm. So each longer side is 9 cm.

Answer:

2

Now, let's solve the kite perimeter problem (assuming the perimeter is 28 cm and shorter sides are 10 cm? Wait, the text is a bit unclear, but let's assume the kite has two pairs of adjacent sides equal. Let the shorter sides be \( a = 10 \) cm (wait, maybe the shorter sides are each 10? No, a kite has two distinct side lengths: let the shorter side length be \( s \) and longer be \( l \). Perimeter \( P = 2s + 2l \). Wait, the text says "perimeter of 28 cm" and "shorter sides is 10 centimeters" (maybe each shorter side is 10? But then 210 + 2l = 28 → 20 + 2l = 28 → 2l = 8 → l = 4? No, that can't be longer. Wait, maybe the shorter side length is 10, but there are two shorter sides (so total for shorter: 210 = 20), then the longer sides: (28 - 20)/2 = 4? No, that's shorter. Wait, maybe the perimeter is 28, and the sum of shorter sides is 10? Wait, the text: "kite with a perimeter of 28 cm, shorter sides is 10 centimeters" – maybe each shorter side is 10? No, that would make perimeter too big. Wait, maybe the shorter side length is \( s \), and there are two shorter sides (so \( 2s = 10 \) → \( s = 5 \))? Wait, the original text is: "kite with a perimeter of 28 cm, shorter sides is 10 centimeters" – maybe a typo, and "shorter sides sum to 10 cm". Let's correct:

Step1: Define variables for kite sides

A kite has two pairs of adjacent sides equal. Let the length of each shorter side be \( s \) and each longer side be \( l \). The perimeter \( P = 2s + 2l \).

Step2: Substitute known values

Assume the sum of the two shorter sides is 10 cm (so \( 2s = 10 \) → \( s = 5 \) cm). Perimeter \( P = 28 \) cm. So \( 28 = 10 + 2l \).

Step3: Solve for \( l \)

Subtract 10 from both sides: \( 2l = 28 - 10 = 18 \). Then divide by 2: \( l = \frac{18}{2} = 9 \) cm. So each longer side is 9 cm.