Question

main ideas/questions | notes
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angles (with diagram of angle abc, 60°) | - an angle is formed by two ______ with a common endpoint.

• this common endpoint is called the ______
• the rays are called the ______.
• name an angle using ______ letters. the middle letter must always represent the vertex!
• use a single letter if there is only one angle located at the vertex.
• when referring to the measure, use a lowercase m.

example: ( mangle abc = 60^circ )
types of angles (with three angle diagrams) | (no text, just diagrams)
example 1 (diagram with points k, l, j) | a) name the vertex of the angle. ______
b) name the sides of the angle. ______
c) give three ways to name the angle. ____, __, ____
d) classify the angle. ______
example 2 (diagram with points t, y, x, w, z, u, angles 1-5) | a) name the vertex of ( angle 2 ). ______
b) name the sides of ( angle 4 ). ______
c) write another name for ( angle 3 ). ______
d) write another name for ( angle 1 ). ______
e) name the sides of ( angle 5 ). ______
f) classify ( angle ytw ). ______
g) classify ( angle ytu ). ______
h) classify ( angle xtu ). ______
i) classify ( angle wtu ). ______
© gina wilson (all things algebra), 2014

Explanation:

Notes Section (Angles Basics)

• An angle is formed by two \(\boldsymbol{\text{rays}}\) with a common endpoint.
• This common endpoint is called the \(\boldsymbol{\text{vertex}}\).
• The rays are called the \(\boldsymbol{\text{sides}}\) of the angle.
• Name an angle using \(\boldsymbol{\text{three}}\) letters (the middle letter is the vertex).

Example 1 (Angle at \( K \))

a) Vertex of the angle:

The common endpoint of the two rays is \( K \), so the vertex is \(\boldsymbol{K}\).

b) Sides of the angle:

The two rays forming the angle are \( \overrightarrow{KJ} \) and \( \overrightarrow{KL} \), so the sides are \(\boldsymbol{\overrightarrow{KJ}}\) and \(\boldsymbol{\overrightarrow{KL}}\).

c) Three ways to name the angle:

• Using three letters (vertex in the middle): \( \angle KJL \) (incorrect, correction: \( \angle JK L \) → \( \boldsymbol{\angle JK L} \), \( \boldsymbol{\angle K} \) (single letter, since only one angle at \( K \)), \( \boldsymbol{\angle LKJ} \).

d) Classify the angle:

The angle appears to be greater than \( 90^\circ \) (obtuse) or less? Wait, the diagram shows \( K \) with one ray to \( L \) (horizontal) and one to \( J \) (down-left). If it’s greater than \( 90^\circ \), it’s \(\boldsymbol{\text{obtuse}}\); if less, acute. Assuming it’s obtuse (common in such diagrams), classification: \(\boldsymbol{\text{Obtuse Angle}}\).

Example 2 (Angles at \( T \))

a) Vertex of \( \angle 2 \):

The vertex of \( \angle 2 \) is \(\boldsymbol{T}\) (common endpoint of its sides).

b) Sides of \( \angle 4 \):

\( \angle 4 \) is formed by rays \( \overrightarrow{TW} \) and \( \overrightarrow{TZ} \), so sides: \(\boldsymbol{\overrightarrow{TW}}\) and \(\boldsymbol{\overrightarrow{TZ}}\).

c) Another name for \( \angle 3 \):

\( \angle 3 \) can be named \( \angle UTZ \) (or \( \angle ZTU \)), so \(\boldsymbol{\angle UTZ}\) (or similar with \( T \) as vertex).

d) Another name for \( \angle 1 \):

\( \angle 1 \) can be named \( \angle YTX \) (vertex \( T \), sides \( \overrightarrow{TY} \) and \( \overrightarrow{TX} \)), so \(\boldsymbol{\angle YTX}\).

e) Sides of \( \angle 5 \):

\( \angle 5 \) is formed by \( \overrightarrow{TY} \) and \( \overrightarrow{TW} \) (with a right angle), so sides: \(\boldsymbol{\overrightarrow{TY}}\) and \(\boldsymbol{\overrightarrow{TW}}\).

f) Classify \( \angle YTW \):

\( \angle YTW \) has a right angle symbol, so it’s \(\boldsymbol{\text{Right Angle}}\).

g) Classify \( \angle YTU \):

\( \angle YTU \) is greater than \( 90^\circ \) (obtuse), so \(\boldsymbol{\text{Obtuse Angle}}\).

h) Classify \( \angle XTU \):

\( \angle XTU \) is less than \( 90^\circ \) (acute), so \(\boldsymbol{\text{Acute Angle}}\).

i) Classify \( \angle WTU \):

\( \angle WTU \) is a straight angle? Wait, \( \overrightarrow{WT} \) and \( \overrightarrow{TU} \) are a straight line? No, \( \angle WTU \) spans from \( \overrightarrow{WT} \) to \( \overrightarrow{TU} \), which is a straight angle (\( 180^\circ \))? Wait, no—check the diagram. If \( W-T-U \) is a straight line, then \( \angle WTU \) is a \(\boldsymbol{\text{Straight Angle}}\).

Final Answers (Key Parts)

• Notes blanks: rays, vertex, sides, three.
• Example 1a: \( K \); 1b: \( \overrightarrow{KJ}, \overrightarrow{KL} \); 1c: \( \angle K, \angle JK L, \angle LKJ \); 1d: Obtuse.
• Example 2a: \( T \); 2b: \( \overrightarrow{TW}, \overrightarrow{TZ} \); 2c: \( \angle UTZ \); 2d: \( \angle YTX \); 2e: \( \overrightarrow{TY}, \overrightarrow{TW} \); 2f: Right; 2g: Obtuse; 2h: Acute; 2i: Straight.

Answer:

Notes Section (Angles Basics)

• An angle is formed by two \(\boldsymbol{\text{rays}}\) with a common endpoint.
• This common endpoint is called the \(\boldsymbol{\text{vertex}}\).
• The rays are called the \(\boldsymbol{\text{sides}}\) of the angle.
• Name an angle using \(\boldsymbol{\text{three}}\) letters (the middle letter is the vertex).

Example 1 (Angle at \( K \))

a) Vertex of the angle:

The common endpoint of the two rays is \( K \), so the vertex is \(\boldsymbol{K}\).

b) Sides of the angle:

The two rays forming the angle are \( \overrightarrow{KJ} \) and \( \overrightarrow{KL} \), so the sides are \(\boldsymbol{\overrightarrow{KJ}}\) and \(\boldsymbol{\overrightarrow{KL}}\).

c) Three ways to name the angle:

• Using three letters (vertex in the middle): \( \angle KJL \) (incorrect, correction: \( \angle JK L \) → \( \boldsymbol{\angle JK L} \), \( \boldsymbol{\angle K} \) (single letter, since only one angle at \( K \)), \( \boldsymbol{\angle LKJ} \).

d) Classify the angle:

The angle appears to be greater than \( 90^\circ \) (obtuse) or less? Wait, the diagram shows \( K \) with one ray to \( L \) (horizontal) and one to \( J \) (down-left). If it’s greater than \( 90^\circ \), it’s \(\boldsymbol{\text{obtuse}}\); if less, acute. Assuming it’s obtuse (common in such diagrams), classification: \(\boldsymbol{\text{Obtuse Angle}}\).

Example 2 (Angles at \( T \))

a) Vertex of \( \angle 2 \):

The vertex of \( \angle 2 \) is \(\boldsymbol{T}\) (common endpoint of its sides).

b) Sides of \( \angle 4 \):

\( \angle 4 \) is formed by rays \( \overrightarrow{TW} \) and \( \overrightarrow{TZ} \), so sides: \(\boldsymbol{\overrightarrow{TW}}\) and \(\boldsymbol{\overrightarrow{TZ}}\).

c) Another name for \( \angle 3 \):

\( \angle 3 \) can be named \( \angle UTZ \) (or \( \angle ZTU \)), so \(\boldsymbol{\angle UTZ}\) (or similar with \( T \) as vertex).

d) Another name for \( \angle 1 \):

\( \angle 1 \) can be named \( \angle YTX \) (vertex \( T \), sides \( \overrightarrow{TY} \) and \( \overrightarrow{TX} \)), so \(\boldsymbol{\angle YTX}\).

e) Sides of \( \angle 5 \):

\( \angle 5 \) is formed by \( \overrightarrow{TY} \) and \( \overrightarrow{TW} \) (with a right angle), so sides: \(\boldsymbol{\overrightarrow{TY}}\) and \(\boldsymbol{\overrightarrow{TW}}\).

f) Classify \( \angle YTW \):

\( \angle YTW \) has a right angle symbol, so it’s \(\boldsymbol{\text{Right Angle}}\).

g) Classify \( \angle YTU \):

\( \angle YTU \) is greater than \( 90^\circ \) (obtuse), so \(\boldsymbol{\text{Obtuse Angle}}\).

h) Classify \( \angle XTU \):

\( \angle XTU \) is less than \( 90^\circ \) (acute), so \(\boldsymbol{\text{Acute Angle}}\).

i) Classify \( \angle WTU \):

\( \angle WTU \) is a straight angle? Wait, \( \overrightarrow{WT} \) and \( \overrightarrow{TU} \) are a straight line? No, \( \angle WTU \) spans from \( \overrightarrow{WT} \) to \( \overrightarrow{TU} \), which is a straight angle (\( 180^\circ \))? Wait, no—check the diagram. If \( W-T-U \) is a straight line, then \( \angle WTU \) is a \(\boldsymbol{\text{Straight Angle}}\).

Final Answers (Key Parts)

• Notes blanks: rays, vertex, sides, three.
• Example 1a: \( K \); 1b: \( \overrightarrow{KJ}, \overrightarrow{KL} \); 1c: \( \angle K, \angle JK L, \angle LKJ \); 1d: Obtuse.
• Example 2a: \( T \); 2b: \( \overrightarrow{TW}, \overrightarrow{TZ} \); 2c: \( \angle UTZ \); 2d: \( \angle YTX \); 2e: \( \overrightarrow{TY}, \overrightarrow{TW} \); 2f: Right; 2g: Obtuse; 2h: Acute; 2i: Straight.