for exercises 24–27, find each length and angle measure for rectangle g h j k. round to the nearest tenth. see examples 3 and 4
24. ( mangle ghk )
25. ( mangle hlj )
26. ( gj )
27. ( hl )
(image of rectangle g h j k with hj = 7, jk = 10, angle at g (∠g) is 52°, diagonals h k and g j intersect at l)

Question

Accepted Answer

### 24. $ m\angle GHK $
# Explanation:
## Step 1: Recall rectangle properties
In rectangle $ G H J K $, $ \angle GHK $ and the given $ \angle G = 52^\circ $ are complementary (since $ \angle GHK + \angle G = 90^\circ $ as triangle $ GHK $ is right - angled at $ H $ because $ G H J K $ is a rectangle).
## Step 2: Calculate $ m\angle GHK $
We know that in a right - triangle, the sum of the two non - right angles is $ 90^\circ $. Let $ m\angle GHK=x $. Then $ x + 52^\circ=90^\circ $.
So, $ x=90^\circ - 52^\circ = 38.0^\circ $ (rounded to the nearest tenth).
# Answer:
$ 38.0^\circ $

### 25. $ m\angle HLJ $
# Explanation:
## Step 1: Recall properties of diagonals in a rectangle
The diagonals of a rectangle are equal and bisect each other. So, $ GL = KL $ and $ HL = JL $. Also, triangle $ GHK $ and triangle $ GHJ $ are congruent. The diagonals of a rectangle intersect each other. The angle $ \angle HLJ $ is the vertical angle to the angle formed by the intersection of the diagonals related to the triangle with angle $ 52^\circ $.
First, in triangle $ GKL $, since $ GL = KL $, triangle $ GKL $ is isosceles with $ \angle G=\angle K = 52^\circ $? No, wait. The diagonals of a rectangle bisect each other, so $ GL = KL $, but the base angles of triangle $ GKL $ are not $ 52^\circ $. Wait, actually, the angle $ \angle HLJ $ is an exterior angle to triangle $ GKL $. The measure of an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
In triangle $ GKL $, $ \angle GLK = 180^\circ- 2\times52^\circ=180 - 104 = 76^\circ $? No, that's wrong. Wait, the correct approach: The diagonals of a rectangle are equal and bisect each other, so $ GL = KL $, and $ \angle G = 52^\circ $, $ \angle GKL=90^\circ - 52^\circ = 38^\circ $? No, let's start over.
In rectangle $ G H J K $, $ GH = 7 $, $ JK = 7 $, $ GK = 10 $? Wait, no, the sides: $ GH = 7 $, $ GK = 10 $ (from the diagram, $ GK = 10 $, $ GH = 7 $). The diagonals $ GJ $ and $ HK $ are equal. The angle $ \angle HLJ $: since the diagonals bisect each other, $ GL = KL $, $ HL = JL $. The angle $ \angle HLJ $ is equal to $ 180^\circ- 2\times(90^\circ - 52^\circ) $? No, better way: The angle $ \angle HLJ $ is equal to $ 2\times52^\circ = 104^\circ $? Wait, no. Let's use the property that the diagonals of a rectangle intersect, and the angle between the diagonals:
In a rectangle, if we consider triangle $ GHK $, $ \angle G = 52^\circ $, $ \angle H = 90^\circ $, $ \angle K=38^\circ $. The diagonals intersect at $ L $. Then $ \angle HLJ $ is equal to $ 180^\circ-(90^\circ - 52^\circ)\times2 $? No, the correct formula: The measure of $ \angle HLJ $ is $ 180^\circ - 2\times(90^\circ - 52^\circ)=180 - 2\times38 = 180 - 76 = 104.0^\circ $ (rounded to the nearest tenth). Wait, let's think again. The angle $ \angle HLJ $ is an exterior angle to triangle $ GHL $. The two non - adjacent interior angles of triangle $ GHL $ are $ \angle G = 52^\circ $ and $ \angle GHL = 90^\circ $? No, $ \angle GHL = 90^\circ $ because $ G H J K $ is a rectangle. Wait, no, $ \angle GHL $ is $ 90^\circ $, $ \angle HGL = 52^\circ $, so the exterior angle $ \angle HLJ=\angle GHL+\angle HGL = 90^\circ+52^\circ = 142^\circ $? No, I'm getting confused. Wait, the correct method:
In rectangle $ G H J K $, $ \angle G = 52^\circ $, $ \angle GHK = 38^\circ $ (from question 24). The diagonals bisect each other, so $ HL = JL $, $ GL = KL $. The angle $ \angle HLJ $ is equal to $ 180^\circ - 2\times38^\circ=180 - 76 = 104^\circ $? No, let's use the law of cosines in triangle $ GHK $ first. In triangle $ GHK $, $ GH = 7 $, $ GK = 10 $, $ HK=\sqrt{7^{2}+10^{2}}=\sqrt{49 + 100}=\sqrt{149}\approx12.2 $. But the diagonals of a rectangle are equal, so $ GJ = HK\approx12.2 $. The diagonals bisect each other, so $ HL=\frac{1}{2}HK $, $ GL=\frac{1}{2}GJ $.
Wait, the angle $ \angle HLJ $: since the diagonals intersect, $ \angle HLJ $ is equal to $ 2\times52^\circ = 104^\circ $? Wait, no. Let's look at the vertical angles. The angle $ \angle HLJ $ and the angle $ \angle GLK $ are vertical angles? No, $ \angle HLJ $ and $ \angle GLK $ are vertical angles. In triangle $ GLK $, $ GL = KL $, so it's isosceles. The base angles of triangle $ GLK $ are $ \angle G=\angle K = 52^\circ $? No, that's not right. Wait, the correct answer is that $ m\angle HLJ = 104.0^\circ $. Let's calculate it as follows:
In triangle $ GHK $, $ \angle G = 52^\circ $, $ \angle H = 90^\circ $, so $ \angle K = 38^\circ $. The diagonals intersect at $ L $, so $ GL = KL $, so triangle $ GLK $ is isosceles with $ \angle GLK=180 - 2\times38=104^\circ $. And $ \angle HLJ=\angle GLK = 104.0^\circ $ (since they are vertical angles).
# Explanation:
## Step 1: Analyze triangle $ GHK $
In rectangle $ G H J K $, $ \angle GHK = 90^\circ $, $ \angle G = 52^\circ $, so $ \angle K=90^\circ - 52^\circ = 38^\circ $.
## Step 2: Analyze triangle $ GLK $
Since the diagonals of a rectangle bisect each other, $ GL = KL $. So triangle $ GLK $ is isosceles with $ \angle G=\angle K = 38^\circ $.
## Step 3: Calculate $ \angle GLK $
Using the angle - sum property of a triangle ($ \angle GLK+ \angle G+\angle K = 180^\circ $), we have $ \angle GLK = 180^\circ-(38^\circ + 38^\circ)=180 - 76 = 104^\circ $.
## Step 4: Determine $ m\angle HLJ $
Since $ \angle HLJ $ and $ \angle GLK $ are vertical angles, $ m\angle HLJ=m\angle GLK = 104.0^\circ $.
# Answer:
$ 104.0^\circ $

### 26. $ GJ $
# Explanation:
## Step 1: Recall the Pythagorean theorem for a rectangle
In a rectangle, the length of the diagonal ($ GJ $) can be found using the Pythagorean theorem. For a rectangle with length $ l $ and width $ w $, the diagonal $ d=\sqrt{l^{2}+w^{2}} $. In rectangle $ G H J K $, $ GH = 7 $ (width) and $ GK = 10 $ (length).
## Step 2: Apply the Pythagorean theorem
Let $ GJ = d $. Then $ d=\sqrt{7^{2}+10^{2}}=\sqrt{49 + 100}=\sqrt{149}\approx12.2 $ (rounded to the nearest tenth).
# Answer:
$ 12.2 $

### 27. $ HL $
# Explanation:
## Step 1: Recall the property of diagonals in a rectangle
The diagonals of a rectangle are equal and bisect each other. So, the length of the diagonal $ HK = GJ\approx12.2 $ (from question 26). And $ HL=\frac{1}{2}HK $ (since the diagonals bisect each other).
## Step 2: Calculate $ HL $
Since $ HK = GJ\approx12.2 $, then $ HL=\frac{1}{2}\times12.2 = 6.1 $ (rounded to the nearest tenth).
# Answer:
$ 6.1 $

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QUESTION IMAGE

Question

Explanation:

24. \( m\angle GHK \)

Step 1: Recall rectangle properties

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

Step 1: Recall properties of diagonals in a rectangle

Step 1: Analyze triangle \( GHK \)

Step 2: Analyze triangle \( GLK \)

Step 3: Calculate \( \angle GLK \)

Step 4: Determine \( m\angle HLJ \)

Answer:

25. \( m\angle HLJ \)