Math / Geometry

QuestionName
GEOMETRY 21: Review for Final Exam
Units 1, 2, 3 (First Semester) Unit 1 - Modeling with Geometry and Definitions (Chapter 1) Unit 2 - Rigid Motions (Chapter 9) $\qquad$ Per. 1,2,4,7.8 Unit 3 - Geometric Relationships and Properties (Chapters 2, 3, 4, 5, 6 ) True or False 1) $\qquad$ Any 2 lines always intersect at one point. 2) $\qquad$ Through any 2 points there is exactly one plane. 3) $\qquad$ Any 3 points are always coplanar. 4) $\qquad$ If $\overline{A B}$ bisects $\overline{C D}$ at point $E$ , then $A E=E B$ .
Use the diagram at right for questions \#5-9. 5) If \Varangle 2 is a right angle and $m \not 4=4 x+10$ degrees, and $m \leqslant 6=8 x-4$ degrees, find $x$ and m $\leqslant 3$ . $x=$ $\qquad$ $\mathrm{m} 43=$ $\qquad$ 6) If $m \nless 6=y$ , then write an expression for the m\&BGF. $\qquad$ 7) If the $\mathrm{m} \times 5=90^{\circ}$ , then name 2 angles that are the complements of $\nless 4$ . $\qquad$ and $\qquad$ 8) If $m \neq 5=90^{\circ}$ , name 2 angles that are supplementary, but do not form a linear pair. $\qquad$ and $\qquad$ 9) $\overline{H J} \perp \overline{F C}$ and $\overline{A D} \perp \overline{F C}$ , then $\overline{A D}$ $\qquad$ HJ
For $\# 10-12$ , identify the type of transformation (translation, reflection, rotation). 10) 11) 12)
For \#13-16, use the following statement: "Linear pairs are supplementary, adjacent angles." 13) Rewrite the statement as a conditional. 14) Write the converse of the conditional. 15) Write the statement as a biconditional. 16) Is the statement a definition? Explain your reasoning.

Studdy Solution

STEP 1

What is this asking? This review covers a mix of true/false questions about basic geometric ideas, solving for angles using angle relationships, identifying transformations, and working with conditional statements related to linear pairs. Watch out! Don't mix up complementary and supplementary angles, and make sure you understand the differences between translations, reflections, and rotations.
Also, be careful with conditional statements and biconditionals!

STEP 2

1. Tackle True/False
2. Angle Relationships and Algebra
3. Identify Transformations
4. Conditional Statements and Definitions

STEP 3

Let's bust these true/false myths!
Two lines can be parallel, so they don't always intersect.
False!

STEP 4

Two points define a unique line, and that line can lie in many planes.
But three non-collinear points define a unique plane.
So, through any two points, there are infinitely many planes, not just one.
False!

STEP 5

Any three points do always lie in the same plane.
We can always find a plane that contains all three.
True!

STEP 6

If $\overline{AB}$ bisects $\overline{CD}$ at $E$ , it means $E$ is the midpoint of $\overline{CD}$ , so $CE = ED$ .
It doesn't say anything about $AE$ and $EB$ .
False!

STEP 7

We're given that $\angle 2$ is a right angle, so it measures $90^\circ$ .
Angles 4 and 6 are vertical angles, so they're equal.
We have $m\angle 4 = 4x + 10$ and $m\angle 6 = 8x - 4$ .
So, $4x + 10 = 8x - 4$ .

STEP 8

Subtracting $4x$ from both sides gives $10 = 4x - 4$ .
Adding 4 to both sides gives $14 = 4x$ .
Dividing both sides by 4 gives $x = \frac{14}{4} = \frac{7}{2} = \textbf{3.5}$ .

STEP 9

Now, $m\angle 4 = 4(3.5) + 10 = 14 + 10 = 24^\circ$ .
Since angles 2 and 4 are complementary, $m\angle 3 = 90^\circ - m\angle 4 = 90^\circ - 24^\circ = \textbf{66}^\circ$ .

STEP 10

If $m\angle 6 = y$ , then $m\angle BGF$ is $180^\circ - y$ because angles 6 and BGF are a linear pair and therefore supplementary.

STEP 11

If $m\angle 5 = 90^\circ$ , then angles complementary to $\angle 4$ are $\angle 3$ and $\angle 1$ , since they each add to $90^\circ$ with $\angle 4$ .

STEP 12

If $m\angle 5 = 90^\circ$ , then angles supplementary to each other but not a linear pair are $\angle 1$ and $\angle 6$ (or $\angle 3$ and $\angle 4$ ).
They add up to $90^\circ + 90^\circ = 180^\circ$ , but they don't form a straight line.

STEP 13

If $\overline{HJ} \perp \overline{FC}$ and $\overline{AD} \perp \overline{FC}$ , then $\overline{AD}$ is parallel to $\overline{HJ}$ because they are both perpendicular to the same line.

STEP 14

Transformation 10 looks like a slide, which is a translation.

STEP 15

Transformation 11 looks like a flip, which is a reflection.

STEP 16

Transformation 12 looks like a turn, which is a rotation.

STEP 17

Conditional statement: If two angles are a linear pair, then they are supplementary and adjacent.

STEP 18

Converse: If two angles are supplementary and adjacent, then they are a linear pair.

STEP 19

Biconditional: Two angles are a linear pair if and only if they are supplementary and adjacent.

STEP 20

Yes, this is a definition.
It gives a clear and concise description of what a linear pair is, using necessary and sufficient conditions.
It works both ways!

STEP 21

1. False
2. False
3. True
4. False
5. $x = 3.5$ , $m\angle 3 = 66^\circ$
6. $180^\circ - y$
7. $\angle 1$ and $\angle 3$
8. $\angle 1$ and $\angle 6$ (or $\angle 3$ and $\angle 4$ )
9. parallel
10. Translation
11. Reflection
12. Rotation
13. If two angles are a linear pair, then they are supplementary and adjacent.
14. If two angles are supplementary and adjacent, then they are a linear pair.
15. Two angles are a linear pair if and only if they are supplementary and adjacent.
16. Yes, it provides necessary and sufficient conditions.

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Studdy Solution

STEP 1

STEP 2

1. Tackle True/False2. Angle Relationships and Algebra3. Identify Transformations4. Conditional Statements and Definitions

STEP 3

Let's **bust these true/false myths**!Two lines *can* be parallel, so they don't *always* intersect.False!

STEP 4

Two points define a unique line, and that line can lie in many planes.But *three* non-collinear points define a unique plane.So, through any *two* points, there are *infinitely many* planes, not just one.False!

STEP 5

Any three points *do* always lie in the same plane.We can always find a plane that contains all three.True!

STEP 6

If AB‾\overline{AB}AB bisects CD‾\overline{CD}CD at EEE, it means EEE is the midpoint of CD‾\overline{CD}CD, so CE=EDCE = EDCE=ED.It *doesn't* say anything about AEAEAE and EBEBEB.False!

STEP 7

We're given that ∠2\angle 2∠2 is a right angle, so it measures 90∘90^\circ90∘.Angles 4 and 6 are vertical angles, so they're equal.We have m∠4=4x+10m\angle 4 = 4x + 10m∠4=4x+10 and m∠6=8x−4m\angle 6 = 8x - 4m∠6=8x−4.So, 4x+10=8x−44x + 10 = 8x - 44x+10=8x−4.

STEP 8

Subtracting 4x4x4x from both sides gives 10=4x−410 = 4x - 410=4x−4.Adding 4 to both sides gives 14=4x14 = 4x14=4x.Dividing both sides by 4 gives x=144=72=3.5x = \frac{14}{4} = \frac{7}{2} = \textbf{3.5}x=414​=27​=3.5.

STEP 9

Now, m∠4=4(3.5)+10=14+10=24∘m\angle 4 = 4(3.5) + 10 = 14 + 10 = 24^\circm∠4=4(3.5)+10=14+10=24∘.Since angles 2 and 4 are complementary, m∠3=90∘−m∠4=90∘−24∘=66∘m\angle 3 = 90^\circ - m\angle 4 = 90^\circ - 24^\circ = \textbf{66}^\circm∠3=90∘−m∠4=90∘−24∘=66∘.

STEP 10

If m∠6=ym\angle 6 = ym∠6=y, then m∠BGFm\angle BGFm∠BGF is 180∘−y180^\circ - y180∘−y because angles 6 and BGF are a linear pair and therefore supplementary.

STEP 11

If m∠5=90∘m\angle 5 = 90^\circm∠5=90∘, then angles complementary to ∠4\angle 4∠4 are ∠3\angle 3∠3 and ∠1\angle 1∠1, since they each add to 90∘90^\circ90∘ with ∠4\angle 4∠4.

STEP 12

STEP 13

If HJ‾⊥FC‾\overline{HJ} \perp \overline{FC}HJ⊥FC and AD‾⊥FC‾\overline{AD} \perp \overline{FC}AD⊥FC, then AD‾\overline{AD}AD is *parallel* to HJ‾\overline{HJ}HJ because they are both perpendicular to the same line.

STEP 14

Transformation 10 looks like a **slide**, which is a translation.

STEP 15

Transformation 11 looks like a **flip**, which is a reflection.

STEP 16

Transformation 12 looks like a **turn**, which is a rotation.

STEP 17

Conditional statement: If two angles are a linear pair, then they are supplementary and adjacent.

STEP 18

Converse: If two angles are supplementary and adjacent, then they are a linear pair.

STEP 19

Biconditional: Two angles are a linear pair if and only if they are supplementary and adjacent.

STEP 20

Yes, this is a definition.It gives a clear and concise description of what a linear pair is, using necessary and sufficient conditions.It works both ways!

STEP 21

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1. Tackle True/False
2. Angle Relationships and Algebra
3. Identify Transformations
4. Conditional Statements and Definitions

Let's bust these true/false myths!
Two lines can be parallel, so they don't always intersect.
False!

Two points define a unique line, and that line can lie in many planes.
But three non-collinear points define a unique plane.
So, through any two points, there are infinitely many planes, not just one.
False!

Any three points do always lie in the same plane.
We can always find a plane that contains all three.
True!

If $\overline{AB}$ bisects $\overline{CD}$ at $E$ , it means $E$ is the midpoint of $\overline{CD}$ , so $CE = ED$ .
It doesn't say anything about $AE$ and $EB$ .
False!

We're given that $\angle 2$ is a right angle, so it measures $90^\circ$ .
Angles 4 and 6 are vertical angles, so they're equal.
We have $m\angle 4 = 4x + 10$ and $m\angle 6 = 8x - 4$ .
So, $4x + 10 = 8x - 4$ .

Subtracting $4x$ from both sides gives $10 = 4x - 4$ .
Adding 4 to both sides gives $14 = 4x$ .
Dividing both sides by 4 gives $x = \frac{14}{4} = \frac{7}{2} = \textbf{3.5}$ .

Now, $m\angle 4 = 4(3.5) + 10 = 14 + 10 = 24^\circ$ .
Since angles 2 and 4 are complementary, $m\angle 3 = 90^\circ - m\angle 4 = 90^\circ - 24^\circ = \textbf{66}^\circ$ .

If $m\angle 6 = y$ , then $m\angle BGF$ is $180^\circ - y$ because angles 6 and BGF are a linear pair and therefore supplementary.

If $m\angle 5 = 90^\circ$ , then angles complementary to $\angle 4$ are $\angle 3$ and $\angle 1$ , since they each add to $90^\circ$ with $\angle 4$ .

If $\overline{HJ} \perp \overline{FC}$ and $\overline{AD} \perp \overline{FC}$ , then $\overline{AD}$ is parallel to $\overline{HJ}$ because they are both perpendicular to the same line.

Transformation 10 looks like a slide, which is a translation.

Transformation 11 looks like a flip, which is a reflection.

Transformation 12 looks like a turn, which is a rotation.

Yes, this is a definition.
It gives a clear and concise description of what a linear pair is, using necessary and sufficient conditions.
It works both ways!