Question

properties of rectangles
color by number
date
directions: answer each question below (diagrams may not be drawn to scale). find your answer in the answer bank and the corresponding color listed next to it. on the next page, write the correct color next to each question number to create your color code. then, color your geometric mandala accordingly
1 which statement is true about rectangles?
all four sides are congruent quad gray
the diagonals are perpendicular quad red
all four angles are congruent quad yellow
2 a parallelogram must be a rectangle when its
diagonals are congruent quad green
opposite sides are parallel quad blue
diagonals bisect the angles quad orange
3 find the perimeter of ( \triangle abc )
diagram of rectangle ( abcd ) with ( ad = 8 ), ( ab = 15 ), right angles at ( d ) and ( b ), diagonal ( ac )
40 quad purple
46 quad yellow
60 quad pink
4 in rectangle ( jklm ), if ( mangle jkn = 33^circ ), find the ( mangle mnl )
diagram of rectangle ( jklm ) with diagonals intersecting at ( n )
33 quad orange
114 quad blue
147 quad gray
5 in rectangle ( abcd ), ( ac = 7x + 12 ) and ( bd = 15x - 8 ). find the length of ( overline{ac} )
25 quad brown
295 quad gray
68 quad red
6 rectangle ( defg ) is shown below. if ( mangle hdg = (9x - 1)^circ ) and ( mangle hgd = (5x + 15)^circ ), find ( mangle hde )
diagram of rectangle ( defg ) with diagonals intersecting at ( h )
55 quad pink
35 quad green
4 quad orange
© acute geometry

Explanation:

Step1: Evaluate rectangle properties

Rectangles have four right (congruent) angles; sides are congruent in pairs, diagonals are congruent but not perpendicular (unless it's a square).

Step2: Identify rectangle condition for parallelograms

A parallelogram is a rectangle if its diagonals are congruent; opposite sides parallel is true for all parallelograms, diagonals bisect angles is true for rhombuses/squares.

Step3: Calculate side AC of $\Delta ABC$

Use Pythagorean theorem: $AC = \sqrt{8^2 + 15^2} = \sqrt{64+225} = \sqrt{289} = 17$

Step4: Find perimeter of $\Delta ABC$

Perimeter = $8 + 15 + 17 = 40$

Step5: Analyze rectangle JKLM angles

In rectangles, diagonals are congruent and bisect each other, so $\triangle JKN$ is isosceles: $\angle JKN = \angle JNK = 33^\circ$. $\angle MJN = 180^\circ - 2(33^\circ) = 114^\circ$. $\angle MNL = \angle MJN$ (vertical angles), so $m\angle MNL = 114^\circ$.

Step6: Solve for x in rectangle ABCD

Rectangle diagonals are congruent: $7x+12 = 15x-8$
$12+8 = 15x-7x$
$20 = 8x$
$x = \frac{20}{8} = 2.5$

Step7: Calculate length of $\overline{AC}$

$AC = 7(2.5)+12 = 17.5 + 12 = 29.5$

Step8: Solve for x in rectangle DEFG

$\angle HDG$ and $\angle HGD$ are complementary (since $\angle DGH$ is part of a right angle in the rectangle):
$9x-1 + 5x+15 = 90$
$14x +14 =90$
$14x=76$
$x=\frac{76}{14}=\frac{38}{7}\approx5.43$

Step9: Find $m\angle HDE$

$\angle HDE = \angle HGD$ (alternate interior angles, since $DE \parallel GF$), so $m\angle HDE = 5(\frac{38}{7})+15 = \frac{190}{7}+15 = \frac{190+105}{7}=\frac{295}{7}\approx42.14$; correcting: $\angle HDE = 90^\circ - \angle HDG = 90-(9x-1)$. Substitute $x=\frac{38}{7}$: $90 - (9(\frac{38}{7})-1)=90 - (\frac{342}{7}-1)=90-\frac{335}{7}=\frac{630-335}{7}=\frac{295}{7}\approx42.14$, but matching the answer bank, the intended solution uses $\angle HDG = \angle HFD$, so $9x-1=5x+15$ (vertical/alternate angles):
$4x=16$, $x=4$. Then $m\angle HDG=9(4)-1=35^\circ$, so $m\angle HDE=90-35=55^\circ$.

Answer:

1. All four angles are congruent (yellow)
2. diagonals are congruent (green)
3. 40 (purple)
4. 114 (blue)
5. 29.5 (gray)
6. 55 (pink)