Question

1. given that ∠efg is a right angle, find ( mangle efh ) and ( mangle hfg ).

diagram: point ( e ) (top), ( f ) (bottom - left, right - angled vertex), ( g ) (bottom - right), ( h ) (on the ray between ( eh ) and ( fg )); ( angle efh = (2x + 2)^circ ), ( angle hfg = (x + 1)^circ ); below the diagram: ( mangle efh = \boxed{34}^circ ), ( mangle hfg = \boxed{57}^circ )

Explanation:

Step1: Set up the equation

Since $\angle EFG$ is a right angle, its measure is $90^\circ$. The sum of $\angle EFH$ and $\angle HFG$ should equal $90^\circ$. So we have the equation:
$$(2x + 2) + (x + 1) = 90$$

Step2: Solve for x

Combine like terms:
$$3x + 3 = 90$$
Subtract 3 from both sides:
$$3x = 87$$
Divide both sides by 3:
$$x = 29$$

Step3: Find $m\angle EFH$

Substitute $x = 29$ into the expression for $\angle EFH$:
$$2x + 2 = 2(29) + 2 = 58 + 2 = 60$$ Wait, but the given answer is 34 and 57. Wait, maybe I misread the angles. Wait, maybe the angles are $(2x + 2)$ and $(x + 1)$ but maybe the sum is 90, but let's check the given answer. Wait, maybe there was a typo, but let's follow the correct steps. Wait, if the given answer is 34 and 57, let's see: 34 + 57 = 91, which is not 90. Wait, maybe the original problem has different expressions. Wait, maybe the angles are $(2x - 2)$ and $(x + 1)$? Let's try: If $2x - 2 + x + 1 = 90$, then $3x -1 =90$, $3x=91$, no. Wait, maybe the angles are $(2x + 2)$ and $(3x - 1)$? No, the diagram shows $(2x + 2)$ and $(x + 1)$. Wait, maybe the user made a mistake in the given answer, but let's do it correctly.

Wait, let's start over. $\angle EFG$ is right, so $m\angle EFH + m\angle HFG = 90^\circ$. So $(2x + 2) + (x + 1) = 90$. So $3x + 3 = 90$, $3x = 87$, $x = 29$. Then $m\angle EFH = 2(29) + 2 = 60^\circ$, $m\angle HFG = 29 + 1 = 30^\circ$. But the given answer is 34 and 57. Wait, maybe the angles are $(2x - 12)$ and $(x + 1)$? Let's try: $2x -12 + x +1 =90$, $3x -11=90$, $3x=101$, no. Wait, maybe the angles are $(2x + 2)$ and $(3x - 1)$? No. Wait, maybe the right angle is 91? No, right angle is 90. Wait, maybe the diagram has different labels. Alternatively, maybe the given answer is wrong, but let's proceed with the correct method.

Wait, perhaps the original problem has $\angle EFG$ as 91 degrees? No, right angle is 90. Alternatively, maybe the expressions are $(2x - 10)$ and $(x + 1)$. Let's try: $2x -10 + x +1 =90$, $3x -9=90$, $3x=99$, $x=33$. Then $2x -10=56$, $x +1=34$. 56 +34=90. No, not 34 and 57. Wait, 34 +57=91. Maybe the right angle is 91? Then $(2x + 2)+(x +1)=91$, $3x +3=91$, $3x=88$, $x=88/3\approx29.33$, not integer. Alternatively, maybe the angles are $(2x + 3)$ and $(x + 4)$. Then $3x +7=90$, $3x=83$, no. Wait, maybe the user made a mistake in the problem statement. But according to the correct method, let's do it:

1. Since $\angle EFG$ is right, $m\angle EFH + m\angle HFG = 90^\circ$.
2. So $(2x + 2) + (x + 1) = 90$.
3. Combine like terms: $3x + 3 = 90$.
4. Subtract 3: $3x = 87$.
5. Divide by 3: $x = 29$.
6. Then $m\angle EFH = 2(29) + 2 = 60^\circ$, $m\angle HFG = 29 + 1 = 30^\circ$.

But the given answer is 34 and 57. Maybe there's a typo in the problem's angle expressions. If we assume the angles are $(2x - 30)$ and $(x + 1)$, then $2x -30 + x +1 =90$, $3x -29=90$, $3x=119$, no. Alternatively, if the angles are $(x + 3)$ and $(2x + 54)$, then $3x +57=90$, $3x=33$, $x=11$. Then $x +3=14$, $2x +54=76$, no. Wait, 34 and 57: 34 is $2x + 2$? Then $2x + 2=34$, $2x=32$, $x=16$. Then $x +1=17$, but 34 +17=51≠90. No. If 57 is $x +1$, then $x=56$, $2x +2=114$, which is more than 90. So there's a mistake. But according to the correct mathematical approach, the steps are as above.

Answer:

To find \( m\angle EFH \) and \( m\angle HFG \) given \( \angle EFG \) is a right angle (\( 90^\circ \)):

Step 1: Set up the angle - sum equation

Since \( \angle EFG=\angle EFH+\angle HFG \) and \( \angle EFG = 90^\circ \), we have:
\[
(2x + 2)+(x + 1)=90
\]

Step 2: Solve for \( x \)

Combine like terms:
\[
3x+3 = 90
\]
Subtract 3 from both sides:
\[
3x=87
\]
Divide by 3:
\[
x = 29
\]

Step 3: Find \( m\angle EFH \) and \( m\angle HFG \)

• For \( m\angle EFH \): Substitute \( x = 29 \) into \( 2x + 2 \):

\[
m\angle EFH=2(29)+2=60^\circ
\]

• For \( m\angle HFG \): Substitute \( x = 29 \) into \( x + 1 \):

\[
m\angle HFG=29 + 1=30^\circ
\]

(Note: The given answer \( 34^\circ \) and \( 57^\circ \) does not satisfy the right - angle condition \( 34 + 57=91
eq90 \). There may be a typo in the problem's angle expressions or the provided answer.)

If we assume there is a typo and the angle expressions are different (e.g., if \( \angle EFH=(2x - 30)^\circ \) and \( \angle HFG=(x + 1)^\circ \) and we force \( 34 + 57 = 91\) as the angle of \( \angle EFG \) (non - standard), but this is not a right angle. The correct mathematical solution for a right - angled \( \angle EFG \) is \( m\angle EFH = 60^\circ \), \( m\angle HFG = 30^\circ \).)