find each measure.
1. ( mangle d )

2. ( mangle l )

for trapezoid ( hjkl ), ( m ) and ( n ) are the midpoints of the legs.
3. if ( hj = 32 ) and ( lk = 60 ), find ( mn ).

exercises
if ( ghjk ) is a kite, find each measure.
1. find ( mangle jrk ).
2. if ( rj = 3 ) and ( rk = 10 ), find ( jk ).
3. if ( mangle ghj = 90 ) and ( mangle gkj = 110 ), find ( mangle hgk ).

Question

find each measure.
1. ( mangle d )

2. ( mangle l )

for trapezoid ( hjkl ), ( m ) and ( n ) are the midpoints of the legs.
3. if ( hj = 32 ) and ( lk = 60 ), find ( mn ).

exercises
if ( ghjk ) is a kite, find each measure.
1. find ( mangle jrk ).
2. if ( rj = 3 ) and ( rk = 10 ), find ( jk ).
3. if ( mangle ghj = 90 ) and ( mangle gkj = 110 ), find ( mangle hgk ).

Accepted Answer

### Problem 1: $ m\angle D $ (Trapezoid $ABCD$)
# Explanation:
## Step 1: Identify Trapezoid Property
In an isosceles trapezoid (or any trapezoid with $AB \parallel CD$), consecutive angles between the bases are supplementary. So $ \angle A + \angle D = 180^\circ $ (since $AB \parallel CD$ and $AD$ is a leg).
## Step 2: Calculate $ m\angle D $
Given $ m\angle A = 125^\circ $, we use $ m\angle D = 180^\circ - m\angle A $.
$ m\angle D = 180^\circ - 125^\circ = 55^\circ $.

# Answer: $ 55^\circ $

### Problem 2: $ m\angle L $ (Kite $JKLM$)
# Explanation:
## Step 1: Identify Kite Property
In a kite, one pair of opposite angles (between the unequal sides) are equal, and consecutive angles between the equal sides are supplementary. Here, $JK = JM = 5$ and $KL = ML$ (implied by markings), so $ \angle K + \angle L = 180^\circ $ (consecutive angles between the equal sides).
## Step 2: Calculate $ m\angle L $
Given $ m\angle K = 40^\circ $, we use $ m\angle L = 180^\circ - m\angle K $.
$ m\angle L = 180^\circ - 40^\circ = 140^\circ $.

# Answer: $ 140^\circ $

### Problem 3: $ MN $ (Trapezoid $ HJKL $, Midsegment)
# Explanation:
## Step 1: Recall Trapezoid Midsegment Formula
The midsegment ($MN$) of a trapezoid is the average of the two bases: $ MN = \frac{HJ + LK}{2} $.
## Step 2: Substitute Values
Given $ HJ = 32 $ and $ LK = 60 $, substitute into the formula:
$ MN = \frac{32 + 60}{2} = \frac{92}{2} = 46 $.

# Answer: $ 46 $

### Exercises (Kite $GHJK$)
#### 1. $ m\angle JRK $
# Explanation:
## Step 1: Kite Diagonals Property
In a kite, the diagonals are perpendicular, so $ \angle JRK = 90^\circ $.

# Answer: $ 90^\circ $

#### 2. $ JK $ (Right Triangle $ JRK $)
# Explanation:
## Step 1: Identify Right Triangle
$ \triangle JRK $ is a right triangle with $ RJ = 3 $, $ RK = 10 $, and $ JK $ as the hypotenuse.
## Step 2: Apply Pythagorean Theorem
$ JK = \sqrt{RJ^2 + RK^2} = \sqrt{3^2 + 10^2} = \sqrt{9 + 100} = \sqrt{109} \approx 10.44 $ (or exact form $ \sqrt{109} $).

# Answer: $ \sqrt{109} $ (or $ \approx 10.44 $)

#### 3. $ m\angle HGK $
# Explanation:
## Step 1: Kite Angle Properties
In a kite, one diagonal bisects the angles. The sum of angles in $ \triangle GHK $: $ m\angle GHJ = 90^\circ $ (so $ m\angle GHR = 45^\circ $), $ m\angle GKJ = 110^\circ $ (so $ m\angle GKH = 55^\circ $).
In $ \triangle HGK $, $ m\angle HGK = 180^\circ - m\angle GHR - m\angle GKH $.
$ m\angle HGK = 180^\circ - 45^\circ - 55^\circ = 80^\circ $.

# Answer: $ 80^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \( m\angle D \) (Trapezoid \(ABCD\))

Step 1: Identify Trapezoid Property

Step 2: Calculate \( m\angle D \)

Step 1: Identify Kite Property

Step 2: Calculate \( m\angle L \)

Step 1: Recall Trapezoid Midsegment Formula

Step 2: Substitute Values

Answer:

Problem 2: \( m\angle L \) (Kite \(JKLM\))