Math / Geometry

QuestionI CAN USE FACTS ABOUT THE EXTERIOR ANGLES OF TRIANGLES TO SOLVE PROBLEMS. 8.8D
20. Four students wrote facts on their dry erase boards about the angle relationships in triangles using the diagram below. Which student wrote a statement that is not true?
CHELSEA BOBBY
KATIE $m \angle 1+m \angle 3=m \angle 5$
MARK $m \angle 2+m \angle 3=m \angle 4$
21. Find the value of $x$ . $x=$ $\qquad$
22. Find the measure of $\angle H T L$ .
23. Find the measure of $\angle C Y L$ .

Studdy Solution

STEP 1

What is this asking? We need to figure out which student's statement about angle relationships in a triangle is false, find the value of $x$ , and find the measures of two angles. Watch out! Don't mix up interior and exterior angles!
Also, remember that diagrams can be deceiving, so stick to the given information.

STEP 2

1. Analyze Student Statements
2. Solve for x
3. Find Angle HTL
4. Find Angle CYL

STEP 3

Let's look at Chelsea's statement: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.
This is the Exterior Angle Theorem, and it's absolutely correct!
So, Chelsea is good!

STEP 4

Now for Bobby: Bobby's statement is also a version of the Exterior Angle Theorem, just using different angles.
Bobby is also correct!

STEP 5

Katie's statement says $m\angle 1 + m\angle 3 = m\angle 5$ . $m\angle 5$ is an exterior angle, and $\angle 1$ and $\angle 3$ are the two nonadjacent interior angles.
This matches the Exterior Angle Theorem, so Katie is right too!

STEP 6

Finally, Mark.
Mark says $m\angle 2 + m\angle 3 = m\angle 4$ . $\angle 4$ is an exterior angle, but $\angle 3$ is adjacent to it, not nonadjacent. $\angle 2$ and $\angle 4$ are also adjacent.
Mark's statement doesn't match the Exterior Angle Theorem, so Mark is incorrect!

STEP 7

We know that $m\angle HTL + m\angle TLY = m\angle HYC$ because of the Exterior Angle Theorem.
We are given $m\angle HTL = (2x + 5)^\circ$ , $m\angle TLY = 55^\circ$ , and $m\angle HYC = 120^\circ$ .

STEP 8

Let's plug those values into our equation: $(2x + 5) + 55 = 120$

STEP 9

Combine like terms: $2x + 60 = 120$

STEP 10

Subtract 60 from both sides: $2x + 60 - 60 = 120 - 60$ $2x = 60$

STEP 11

Divide both sides by 2: $\frac{2x}{2} = \frac{60}{2}$ $x = 30$

STEP 12

We know $m\angle HTL = (2x + 5)^\circ$ and we found $x = 30$ , so let's substitute: $m\angle HTL = (2 \cdot 30 + 5)^\circ$

STEP 13

Calculate: $m\angle HTL = (60 + 5)^\circ$ $m\angle HTL = 65^\circ$

STEP 14

We know that the angles in a triangle add up to $180^\circ$ .
In $\triangle CYL$ , we have $m\angle YCL = 55^\circ$ and $m\angle CLY = 120^\circ$ is given.
So, $m\angle CYL + 55 + 120 = 180$ .

STEP 15

Combine like terms: $m\angle CYL + 175 = 180$

STEP 16

Subtract 175 from both sides: $m\angle CYL + 175 - 175 = 180 - 175$ $m\angle CYL = 5^\circ$

STEP 17

Mark's statement is not true. $x = 30$ . $m\angle HTL = 65^\circ$ . $m\angle CYL = 5^\circ$ .

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Studdy Solution

STEP 1

STEP 2

1. Analyze Student Statements
2. Solve for x
3. Find Angle HTL
4. Find Angle CYL

STEP 3

Let's look at Chelsea's statement: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.
This is the Exterior Angle Theorem, and it's absolutely correct!
So, Chelsea is good!

STEP 4

Now for Bobby: Bobby's statement is also a version of the Exterior Angle Theorem, just using different angles.
Bobby is also correct!

STEP 5

Katie's statement says $m\angle 1 + m\angle 3 = m\angle 5$ . $m\angle 5$ is an exterior angle, and $\angle 1$ and $\angle 3$ are the two nonadjacent interior angles.
This matches the Exterior Angle Theorem, so Katie is right too!

STEP 6

Finally, Mark.
Mark says $m\angle 2 + m\angle 3 = m\angle 4$ . $\angle 4$ is an exterior angle, but $\angle 3$ is adjacent to it, not nonadjacent. $\angle 2$ and $\angle 4$ are also adjacent.
Mark's statement doesn't match the Exterior Angle Theorem, so Mark is incorrect!

STEP 7

We know that $m\angle HTL + m\angle TLY = m\angle HYC$ because of the Exterior Angle Theorem.
We are given $m\angle HTL = (2x + 5)^\circ$ , $m\angle TLY = 55^\circ$ , and $m\angle HYC = 120^\circ$ .

STEP 8

Let's plug those values into our equation: $(2x + 5) + 55 = 120$

STEP 9

Combine like terms: $2x + 60 = 120$

STEP 10

Subtract 60 from both sides: $2x + 60 - 60 = 120 - 60$ $2x = 60$

STEP 11

Divide both sides by 2: $\frac{2x}{2} = \frac{60}{2}$ $x = 30$

STEP 12

We know $m\angle HTL = (2x + 5)^\circ$ and we found $x = 30$ , so let's substitute: $m\angle HTL = (2 \cdot 30 + 5)^\circ$

STEP 13

Calculate: $m\angle HTL = (60 + 5)^\circ$ $m\angle HTL = 65^\circ$

STEP 14

We know that the angles in a triangle add up to $180^\circ$ .
In $\triangle CYL$ , we have $m\angle YCL = 55^\circ$ and $m\angle CLY = 120^\circ$ is given.
So, $m\angle CYL + 55 + 120 = 180$ .

STEP 15

Combine like terms: $m\angle CYL + 175 = 180$

STEP 16

Subtract 175 from both sides: $m\angle CYL + 175 - 175 = 180 - 175$ $m\angle CYL = 5^\circ$

STEP 17

Mark's statement is not true. $x = 30$ . $m\angle HTL = 65^\circ$ . $m\angle CYL = 5^\circ$ .

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Studdy Solution

STEP 1

STEP 2

1. Analyze Student Statements2. Solve for x3. Find Angle HTL4. Find Angle CYL

STEP 3

Let's look at **Chelsea's** statement: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.This is the **Exterior Angle Theorem**, and it's absolutely correct!So, Chelsea is good!

STEP 4

Now for **Bobby**: Bobby's statement is also a version of the **Exterior Angle Theorem**, just using different angles.Bobby is also correct!

STEP 5

**Katie's** statement says m∠1+m∠3=m∠5m\angle 1 + m\angle 3 = m\angle 5m∠1+m∠3=m∠5. m∠5m\angle 5m∠5 is an exterior angle, and ∠1\angle 1∠1 and ∠3\angle 3∠3 are the two nonadjacent interior angles.This matches the **Exterior Angle Theorem**, so Katie is right too!

STEP 6

STEP 7

STEP 8

Let's plug those values into our equation: (2x+5)+55=120(2x + 5) + 55 = 120(2x+5)+55=120

STEP 9

Combine like terms: 2x+60=1202x + 60 = 1202x+60=120

STEP 10

Subtract 60 from both sides: 2x+60−60=120−602x + 60 - 60 = 120 - 602x+60−60=120−60 2x=602x = 602x=60

STEP 11

Divide both sides by 2: 2x2=602\frac{2x}{2} = \frac{60}{2}22x​=260​ x=30x = 30x=30

STEP 12

We know m∠HTL=(2x+5)∘m\angle HTL = (2x + 5)^\circm∠HTL=(2x+5)∘ and we found x=30x = 30x=30, so let's substitute: m∠HTL=(2⋅30+5)∘m\angle HTL = (2 \cdot 30 + 5)^\circm∠HTL=(2⋅30+5)∘

STEP 13

Calculate: m∠HTL=(60+5)∘m\angle HTL = (60 + 5)^\circm∠HTL=(60+5)∘ m∠HTL=65∘m\angle HTL = 65^\circm∠HTL=65∘

STEP 14

We know that the angles in a triangle add up to 180∘180^\circ180∘.In △CYL\triangle CYL△CYL, we have m∠YCL=55∘m\angle YCL = 55^\circm∠YCL=55∘ and m∠CLY=120∘m\angle CLY = 120^\circm∠CLY=120∘ is given.So, m∠CYL+55+120=180m\angle CYL + 55 + 120 = 180m∠CYL+55+120=180.

STEP 15

Combine like terms: m∠CYL+175=180m\angle CYL + 175 = 180m∠CYL+175=180

STEP 16

Subtract 175 from both sides: m∠CYL+175−175=180−175m\angle CYL + 175 - 175 = 180 - 175m∠CYL+175−175=180−175 m∠CYL=5∘m\angle CYL = 5^\circm∠CYL=5∘

STEP 17

Mark's statement is not true. x=30x = 30x=30. m∠HTL=65∘m\angle HTL = 65^\circm∠HTL=65∘. m∠CYL=5∘m\angle CYL = 5^\circm∠CYL=5∘.

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1. Analyze Student Statements
2. Solve for x
3. Find Angle HTL
4. Find Angle CYL

Let's look at Chelsea's statement: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.
This is the Exterior Angle Theorem, and it's absolutely correct!
So, Chelsea is good!

Now for Bobby: Bobby's statement is also a version of the Exterior Angle Theorem, just using different angles.
Bobby is also correct!

Katie's statement says $m\angle 1 + m\angle 3 = m\angle 5$ . $m\angle 5$ is an exterior angle, and $\angle 1$ and $\angle 3$ are the two nonadjacent interior angles.
This matches the Exterior Angle Theorem, so Katie is right too!

Let's plug those values into our equation: $(2x + 5) + 55 = 120$

Combine like terms: $2x + 60 = 120$

Subtract 60 from both sides: $2x + 60 - 60 = 120 - 60$ $2x = 60$

Divide both sides by 2: $\frac{2x}{2} = \frac{60}{2}$ $x = 30$

We know $m\angle HTL = (2x + 5)^\circ$ and we found $x = 30$ , so let's substitute: $m\angle HTL = (2 \cdot 30 + 5)^\circ$

Calculate: $m\angle HTL = (60 + 5)^\circ$ $m\angle HTL = 65^\circ$

We know that the angles in a triangle add up to $180^\circ$ .
In $\triangle CYL$ , we have $m\angle YCL = 55^\circ$ and $m\angle CLY = 120^\circ$ is given.
So, $m\angle CYL + 55 + 120 = 180$ .

Combine like terms: $m\angle CYL + 175 = 180$

Subtract 175 from both sides: $m\angle CYL + 175 - 175 = 180 - 175$ $m\angle CYL = 5^\circ$

Mark's statement is not true. $x = 30$ . $m\angle HTL = 65^\circ$ . $m\angle CYL = 5^\circ$ .