Math  /  Geometry

QuestionUse the diagram and the given information to answer parts (a)-(d). - ADundefined\overleftrightarrow{A D} and EIundefined\overleftrightarrow{E I} are parallel. - JPundefined\overleftrightarrow{J P} and KOundefined\overleftrightarrow{K O} are transversals. - The measure of BCQ\angle B C Q is 6767^{\circ}. - The measure of QHI\angle Q H I is 119119^{\circ}. a. Find the measure of QFH\angle Q F H. b. What is the angle relationship between BCQ\angle B C Q and QFH\angle Q F H that verifies the measure of QFH\angle Q F H ? c. Find the measure of FQH\angle F Q H. d. What is the relationship between FQH,QFH\angle F Q H, \angle Q F H, and QHI\angle Q H I that verifies the measure of FQH\angle F Q H ? Copyri

Studdy Solution

STEP 1

What is this asking? We're looking for the measures of angles within a set of intersecting lines, specifically focusing on the relationships between angles formed by parallel lines and transversals. Watch out! Don't mix up the different types of angle relationships (corresponding, alternate interior, alternate exterior, etc.)!

STEP 2

1. Find the measure of QFH\angle QFH.
2. Determine the angle relationship between BCQ\angle BCQ and QFH\angle QFH.
3. Find the measure of FQH\angle FQH.
4. Determine the relationship between FQH\angle FQH, QFH\angle QFH, and QHI\angle QHI.

STEP 3

We know that ADundefined\overleftrightarrow{AD} and EIundefined\overleftrightarrow{EI} are parallel, and JPundefined\overleftrightarrow{JP} is a transversal.
This makes BCQ\angle BCQ and QFH\angle QFH *corresponding angles*.

STEP 4

Corresponding angles are **congruent**, meaning they have the same measure.
Since BCQ\angle BCQ measures 67\boldsymbol{67^\circ}, QFH\angle QFH also measures 67\boldsymbol{67^\circ}!

STEP 5

As we just saw, BCQ\angle BCQ and QFH\angle QFH are **corresponding angles**.
They are in matching positions relative to the parallel lines and the transversal.

STEP 6

Notice that QFH\angle QFH, FQH\angle FQH, and QHI\angle QHI form a straight line.
This means they are **supplementary angles**, and their measures add up to 180\boldsymbol{180^\circ}.

STEP 7

We can write this relationship as: QFH+FQH+QHI=180 \angle QFH + \angle FQH + \angle QHI = 180^\circ

STEP 8

We know QFH=67\angle QFH = 67^\circ and QHI=119\angle QHI = 119^\circ.
Let's substitute those values: 67+FQH+119=180 67^\circ + \angle FQH + 119^\circ = 180^\circ

STEP 9

Combine the known angle measures: 186+FQH=180 186^\circ + \angle FQH = 180^\circ

STEP 10

To isolate FQH\angle FQH, we subtract 186186^\circ from both sides of the equation: FQH=180186 \angle FQH = 180^\circ - 186^\circ

STEP 11

Therefore, FQH=6\angle FQH = \boldsymbol{-6^\circ}.
Wait a minute... a negative angle?
That doesn't make sense in this context!
Let's go back and check our work.

STEP 12

Aha! It looks like we made a mistake in 2.3.1. QFH\angle QFH, FQH\angle FQH, and QHI\angle QHI *don't* form a straight line. FQH\angle FQH and QHI\angle QHI are adjacent angles, and QFH\angle QFH and FQH\angle FQH are two angles in a triangle.

STEP 13

Since QFH\angle QFH and QHI\angle QHI are same-side interior angles, we know that QFH+QHI=180\angle QFH + \angle QHI = 180^\circ.
Since QHI=119\angle QHI = 119^\circ, we can find QFH=180119=61\angle QFH = 180^\circ - 119^\circ = 61^\circ.

STEP 14

Now, we know two angles in FQH\triangle FQH: QFH=61\angle QFH = 61^\circ and QHF=119\angle QHF = 119^\circ.
Since the angles in a triangle add up to 180180^\circ, we have 61+119+FQH=18061^\circ + 119^\circ + \angle FQH = 180^\circ, so 180+FQH=180180^\circ + \angle FQH = 180^\circ, and FQH=0\angle FQH = \boldsymbol{0^\circ}.
This still doesn't seem right.

STEP 15

Let's reconsider our approach.
We know QFH=67\angle QFH = 67^\circ.
We also know that QHI=119\angle QHI = 119^\circ.
Angles QHF\angle QHF and QHI\angle QHI form a straight line, so they are supplementary, meaning QHF+QHI=180\angle QHF + \angle QHI = 180^\circ.
Thus, QHF=180119=61\angle QHF = 180^\circ - 119^\circ = 61^\circ.
Now, in FQH\triangle FQH, we have QFH=67\angle QFH = 67^\circ and QHF=61\angle QHF = 61^\circ.
Since the angles in a triangle add up to 180180^\circ, we have 67+61+FQH=18067^\circ + 61^\circ + \angle FQH = 180^\circ, so 128+FQH=180128^\circ + \angle FQH = 180^\circ.
Therefore, FQH=180128=52\angle FQH = 180^\circ - 128^\circ = \boldsymbol{52^\circ}.

STEP 16

FQH\angle FQH, QFH\angle QFH, and QHF\angle QHF are the **interior angles of triangle** FQH\triangle FQH.
The sum of the interior angles of any triangle is always 180\boldsymbol{180^\circ}. QHI\angle QHI is supplementary to QHF\angle QHF.

STEP 17

a. The measure of QFH\angle QFH is 67\boldsymbol{67^\circ}. b. BCQ\angle BCQ and QFH\angle QFH are **corresponding angles**. c. The measure of FQH\angle FQH is 52\boldsymbol{52^\circ}. d. FQH\angle FQH, QFH\angle QFH, and QHF\angle QHF are the interior angles of FQH\triangle FQH, and their measures add up to 180180^\circ. QHI\angle QHI is supplementary to QHF\angle QHF.

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