5.1 Perpendicular and Angle Bisectors Name | Schemes and Mind Maps ... - Free Printable

Let's go through the worksheet "5.1 Perpendicular and Angle Bisectors" step by step, filling in the blanks and solving each part with explanations.

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Review:

#### Perpendicular Bisector:
> A line perpendicular to a segment at its midpoint.

✔ *Explanation:* A perpendicular bisector intersects a segment at a 90° angle (perpendicular) and passes through the exact middle (midpoint) of the segment.

- You try: Draw $\overline{OK}$. Draw line $M$ that is the perpendicular bisector of $\overline{OK}$.
- How to do it: Find the midpoint of segment $\overline{OK}$, then draw a line through that point that forms a right angle with $\overline{OK}$.

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#### Angle Bisector:
> A ray that divides an angle into two congruent angles.

✔ *Explanation:* An angle bisector splits an angle into two equal parts. The bisector is usually a ray starting from the vertex of the angle.

- You try: Draw $\angle ABC$. Draw $\overline{BX}$ that is the angle bisector.
- How to do it: Draw angle $ABC$, then draw a ray from point $B$ inside the angle such that it divides $\angle ABC$ into two equal angles.

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#### Equidistant:
> When a point is the same distance from two or more objects.

✔ *Explanation:* For example, any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.

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Angle Review:

We are given a diagram with rays $\overrightarrow{WX}$, $\overrightarrow{WY}$, and $\overrightarrow{WZ}$ all starting from point $W$.

#### 1) Name three angles in the picture:

✔ Possible answers:
- $\angle XWY$
- $\angle YWZ$
- $\angle XWZ$

*(Any three angles formed by these rays.)*

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#### 2) If $\overrightarrow{WY}$ bisects $\angle XWZ$, then which two angles are congruent?

✔ Answer: $\angle XWY \cong \angle YWZ$

*Explanation:* Since $\overrightarrow{WY}$ is the bisector, it divides $\angle XWZ$ into two equal parts.

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#### 3) Draw obtuse $\angle CAT$. Draw the angle bisector $\overrightarrow{AS}$. Which two angles are congruent?

✔ Step-by-step:
- Draw point $C$, $A$, and $T$ so that $\angle CAT$ is greater than 90° (obtuse).
- From point $A$, draw a ray $\overrightarrow{AS}$ that splits $\angle CAT$ into two equal parts.
- Then: $\angle CAS \cong \angle SAT$

✔ Answer: $\angle CAS \cong \angle SAT$

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#### 4) Name the numbered angles:

Given a parallelogram $ABCD$ with diagonal $BD$, forming four angles labeled:

- ∠1: at vertex $A$, between sides $AD$ and $AB$
- ∠2: at vertex $B$, between sides $AB$ and $BD$
- ∠3: at vertex $B$, between sides $BD$ and $BC$
- ∠4: at vertex $D$, between sides $DA$ and $DB$

So:

- ∠1 = $\angle DAB$
- ∠2 = $\angle ABD$
- ∠3 = $\angle CBD$
- ∠4 = $\angle ADB$

✔ Answers:
- 1: $\angle DAB$
- 2: $\angle ABD$
- 3: $\angle CBD$
- 4: $\angle ADB$

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Final Note:

> *The middle letter of an angle represents the ________!*

✔ Answer: Vertex

*Explanation:* In $\angle ABC$, point $B$ is the vertex — the common endpoint of the two rays forming the angle.

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✔ Final Completed Worksheet Summary:

#### Review:
- Perpendicular Bisector: A line perpendicular to a segment at its midpoint.
- Angle Bisector: A ray that divides an angle into two congruent angles.
- Equidistant: When a point is the same distance from two or more objects.

#### Angle Review:
1) Three angles: $\angle XWY$, $\angle YWZ$, $\angle XWZ$
2) $\angle XWY \cong \angle YWZ$
3) $\angle CAS \cong \angle SAT$
4)
- 1: $\angle DAB$
- 2: $\angle ABD$
- 3: $\angle CBD$
- 4: $\angle ADB$

> *The middle letter of an angle represents the vertex!*

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