# Geometry

## Educators MB

Problem 1

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.)
Given: Point $Y$ lies between points $X$ and $Z$ .
$\begin{array}{ll}{\text { a. } X Y=\frac{1}{2} X Z} & {\text { b. } X Z=X Y+Y Z} \\ {\text { c. } X Z>X Y} & {\text { d. } Y Z>X Y} \\ {\text { e. } X Z>Y Z} & {\text { f. } X Z>2 X Y}\end{array}$ Amrita B.

Problem 2

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.)

Given: Point $B$ lies in the interior of $\angle A O C$ .
$\begin{array}{ll}{\text { a. } m \angle 1=m \angle 2} & {\text { b. } m \angle A O C=m \angle 1+m \angle 2} \\ {\text { c. } m \angle A O C>m \angle 1} & {\text { d. } m \angle A O C>m \angle 2} \\ {\text { e. } m \angle 1>m \angle 2} & {\text { f. } m \angle A O C>90}\end{array}$

MB
Manav B.

Problem 3

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.)

Given: $\square A B C D ; A C>B D$
$\begin{array}{ll}{\text { a. } A B>A D} & {\text { b. } A M>M C} \\ {\text { c. } D M=M B} & {\text { d. } A M>M B}\end{array}$

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Problem 4

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.)

Given: $m \angle R V S=m \angle R S V=65$
$\begin{array}{ll}{\text { a. } R T > R S} & {\text { b. } R T > R V} \\ {\text { c. } R S > S T} & {\text { d. } V T< R S}\end{array}$

MB
Manav B.

Problem 5

When some people are given that $j>k$ and $l>m$ , they carelessly conclude that $j+k>l+m .$ Find values for $j, k, l$ , and $m$ that show this conclusion is false. Amrita B.

Problem 6

Write the reasons that justify the statements.
Given: $\triangle A B C \cong \triangle R S T$
Prove: $A K>R S$
1. $\triangle A B C \cong \triangle R S T$
2. $\overline{A B} \cong \overline{R S},$ or $A B=R S$
3. $A K=A B+B K$
4. $A K>A B$
5. $A K>R S$

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Problem 7

Write the reasons that justify the statements.
Given: $\overrightarrow{D E}, \overrightarrow{F G}$ and $\overrightarrow{Z H}$ contain point $Z$ .
Prove: $m \angle D Z H>m \angle G Z E$
1. $\angle D Z F \cong \angle G Z E$ . or $m \angle D Z F=m \angle G Z E$
2. $m \angle D Z H=m \angle D Z F+m \angle F Z H$
3. $m \angle D Z H>m \angle D Z F$
4. $m \angle D Z H>m \angle G Z E$ Amrita B.

Problem 8

Write proofs in two-column form.
Given: $K L>N L ; L M>L P$
Prove: $K M>N P$

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Problem 9

Write proofs in two-column form.
Given: $m \angle R O S>m \angle T O V$
Prove: $m \angle R O T>m \angle S O V$ Amrita B.

Problem 10

Write proofs in two-column form.
Given: $\overline{V Y} \perp \overline{Y Z}$
Prove: $\angle V X Z$ is an obtuse angle.

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Problem 11

Write proofs in two-column form.
Given: The diagram
Prove: $m \angle 1>m \angle 4$ Amrita B.

Problem 12

Write proofs in two-column form.
Given: $\overline{Q R}$ and $\overline{S T}$ bisect each other.
Prove: $m \angle X R T>m \angle S$

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Problem 13

Write proofs in two-column form.
Given: Point $K$ lies inside $\triangle A B C$ .
Prove: $m \angle K>m \angle C$ Amrita B.