Math / Geometry

QuestionFind the measures of angles $x, y$ , and $z$ in the figure. m||n. $\angle x=$ $\square$

Studdy Solution

STEP 1

1. Lines $m$ and $n$ are parallel.
2. The transversal creates corresponding, alternate interior, and consecutive interior angles.
3. The given angle adjacent to $x$ is $76^\circ$ .

STEP 2

1. Identify the relationship between angle $x$ and the given angle.
2. Use properties of parallel lines to find angle $y$ .
3. Use properties of parallel lines to find angle $z$ .

STEP 3

Since angle $x$ is adjacent to the given angle of $76^\circ$ , they form a linear pair. Linear pairs are supplementary, meaning their measures add up to $180^\circ$ .

STEP 4

Calculate angle $x$ using the supplementary angle relationship:
$x + 76^\circ = 180^\circ$

STEP 5

Solve for $x$ :
$x = 180^\circ - 76^\circ$ $x = 104^\circ$

STEP 6

Since lines $m$ and $n$ are parallel, angle $y$ is a corresponding angle to the given angle of $76^\circ$ . Corresponding angles are equal.
$y = 76^\circ$

STEP 7

Angle $z$ is an alternate interior angle to angle $x$ . Alternate interior angles are equal when lines are parallel.
$z = x = 104^\circ$
The measures of the angles are: $\angle x = 104^\circ$ $\angle y = 76^\circ$ $\angle z = 104^\circ$

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QuestionFind the measures of angles $x, y$ , and $z$ in the figure. m||n. $\angle x=$ $\square$

Studdy Solution

STEP 1

1. Lines $m$ and $n$ are parallel.
2. The transversal creates corresponding, alternate interior, and consecutive interior angles.
3. The given angle adjacent to $x$ is $76^\circ$ .

STEP 2

1. Identify the relationship between angle $x$ and the given angle.
2. Use properties of parallel lines to find angle $y$ .
3. Use properties of parallel lines to find angle $z$ .

STEP 3

Since angle $x$ is adjacent to the given angle of $76^\circ$ , they form a linear pair. Linear pairs are supplementary, meaning their measures add up to $180^\circ$ .

STEP 4

Calculate angle $x$ using the supplementary angle relationship:
$x + 76^\circ = 180^\circ$

STEP 5

Solve for $x$ :
$x = 180^\circ - 76^\circ$ $x = 104^\circ$

STEP 6

Since lines $m$ and $n$ are parallel, angle $y$ is a corresponding angle to the given angle of $76^\circ$ . Corresponding angles are equal.
$y = 76^\circ$

STEP 7

Angle $z$ is an alternate interior angle to angle $x$ . Alternate interior angles are equal when lines are parallel.
$z = x = 104^\circ$
The measures of the angles are: $\angle x = 104^\circ$ $\angle y = 76^\circ$ $\angle z = 104^\circ$

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QuestionFind the measures of angles x,yx, yx,y, and zzz in the figure. m||n. ∠x=\angle x=∠x= □\square□

Studdy Solution

STEP 1

1. Lines m m m and n n n are parallel.2. The transversal creates corresponding, alternate interior, and consecutive interior angles.3. The given angle adjacent to x x x is 76∘ 76^\circ 76∘.

STEP 2

1. Identify the relationship between angle x x x and the given angle.2. Use properties of parallel lines to find angle y y y.3. Use properties of parallel lines to find angle z z z.

STEP 3

Since angle x x x is adjacent to the given angle of 76∘ 76^\circ 76∘, they form a linear pair. Linear pairs are supplementary, meaning their measures add up to 180∘ 180^\circ 180∘.

STEP 4

Calculate angle x x x using the supplementary angle relationship:x+76∘=180∘ x + 76^\circ = 180^\circ x+76∘=180∘

STEP 5

Solve for x x x:x=180∘−76∘ x = 180^\circ - 76^\circ x=180∘−76∘ x=104∘ x = 104^\circ x=104∘

STEP 6

Since lines m m m and n n n are parallel, angle y y y is a corresponding angle to the given angle of 76∘ 76^\circ 76∘. Corresponding angles are equal.y=76∘ y = 76^\circ y=76∘

STEP 7

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QuestionFind the measures of angles $x, y$ , and $z$ in the figure. m||n. $\angle x=$ $\square$

1. Lines $m$ and $n$ are parallel.
2. The transversal creates corresponding, alternate interior, and consecutive interior angles.
3. The given angle adjacent to $x$ is $76^\circ$ .

1. Identify the relationship between angle $x$ and the given angle.
2. Use properties of parallel lines to find angle $y$ .
3. Use properties of parallel lines to find angle $z$ .

Since angle $x$ is adjacent to the given angle of $76^\circ$ , they form a linear pair. Linear pairs are supplementary, meaning their measures add up to $180^\circ$ .

Calculate angle $x$ using the supplementary angle relationship:
$x + 76^\circ = 180^\circ$

Solve for $x$ :
$x = 180^\circ - 76^\circ$ $x = 104^\circ$

Since lines $m$ and $n$ are parallel, angle $y$ is a corresponding angle to the given angle of $76^\circ$ . Corresponding angles are equal.
$y = 76^\circ$