Mix Examples - Lines and Angles Questions in English

(A) In any triangle $\Delta ABC$,the sum of all interior angles is $180^{\circ}$.
Therefore,$\angle A + \angle B + \angle C = 180^{\circ}$.
Given that $\angle A = \angle B + \angle C$.
Substituting $(\angle B + \angle C)$ with $\angle A$ in the sum equation:
$\angle A + (\angle B + \angle C) = 180^{\circ}$
$\angle A + \angle A = 180^{\circ}$
$2\angle A = 180^{\circ}$
$\angle A = \frac{180^{\circ}}{2} = 90^{\circ}$.

(B) In any triangle $\Delta ABC$,the sum of all interior angles is $180^{\circ}$.
Therefore,$\angle A + \angle B + \angle C = 180^{\circ}$.
Given that $\angle A = \angle B = \angle C$,let each angle be $x$.
Substituting this into the equation,we get $x + x + x = 180^{\circ}$.
$3x = 180^{\circ}$.
$x = 180^{\circ} / 3 = 60^{\circ}$.
Thus,$\angle A = \angle B = \angle C = 60^{\circ}$.

(C) According to the Exterior Angle Theorem of a triangle,the measure of an exterior angle is equal to the sum of the two interior opposite angles.
In $\Delta PQR$,$\angle PRT$ is the exterior angle at vertex $R$.
The interior opposite angles are $\angle P$ and $\angle Q$.
Therefore,$\angle PRT = \angle P + \angle Q$.
Given $\angle P = 70^{\circ}$ and $\angle Q = 50^{\circ}$.
$\angle PRT = 70^{\circ} + 50^{\circ} = 120^{\circ}$.

(D) Let the acute angle be $x$.
By definition,the complementary angle of $x$ is $(90^{\circ} - x)$.
The supplementary angle of $x$ is $(180^{\circ} - x)$.
The difference between the supplementary angle and the complementary angle is:
$(180^{\circ} - x) - (90^{\circ} - x)$
$= 180^{\circ} - x - 90^{\circ} + x$
$= 180^{\circ} - 90^{\circ}$
$= 90^{\circ}$
Therefore,the difference is always $90^{\circ}$.

(A) According to the Exterior Angle Theorem of a triangle,the measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.
In $\Delta ABC$,$\angle ACD$ is the exterior angle at vertex $C$.
Therefore,$\angle ACD = \angle A + \angle B$.
Given that $\angle ACD = 122^{\circ}$ and $\angle A = 68^{\circ}$.
Substituting the values,we get: $122^{\circ} = 68^{\circ} + \angle B$.
Solving for $\angle B$: $\angle B = 122^{\circ} - 68^{\circ} = 54^{\circ}$.
Thus,$\angle B = 54^{\circ}$.

(C) line is a collection of points that extends infinitely in both directions. $A$ part of a line that has two distinct end points is known as a line segment. Therefore,the correct answer is a line segment.

(B) line is a straight path that extends infinitely in both directions. $A$ line segment is a part of a line with two end points. $A$ ray is a part of a line that starts at one point (the end point) and extends infinitely in one direction. Therefore,a part of a line with one end point is called a ray.

(D) An angle whose measure is exactly $180^{\circ}$ is called a straight angle. This angle represents a straight line.

(B) An angle that measures exactly $90^{\circ}$ is defined as a right angle. Therefore,the correct answer is a right angle.

(C) An angle whose measure is greater than $180^{\circ}$ and less than $360^{\circ}$ is known as a reflex angle.

(C) linear pair of angles is formed when two adjacent angles are created by the intersection of two lines,such that their non-common sides form a straight line.
By the linear pair axiom,if a ray stands on a line,then the sum of two adjacent angles so formed is $180^{\circ}$.
Therefore,the sum of a linear pair of angles is always $180^{\circ}$.

(A) When two lines intersect at a point,they form two pairs of vertically opposite angles. According to the Vertically Opposite Angles Theorem,these angles are always equal to each other.

(B) When two parallel lines are intersected by a transversal,the interior angles on the same side of the transversal are consecutive interior angles (also known as co-interior angles).
According to the properties of parallel lines,these angles are supplementary,meaning their sum is equal to $180^{\circ}$.

(B) According to the angle sum property of a triangle,the sum of all interior angles of any triangle is always equal to $180^o$.
Therefore,the sum of the three angles of a triangle is $180^o$.

(A) When two parallel lines are intersected by a transversal,the corresponding angles formed at each intersection are equal. This is a fundamental property of parallel lines known as the Corresponding Angles Axiom.

(D) In $\Delta ABC$,the exterior angle at vertex $B$ is $\angle ABD = 140^{\circ}$. Since the sum of an interior angle and its adjacent exterior angle is $180^{\circ}$,we have $\angle ABC = 180^{\circ} - 140^{\circ} = 40^{\circ}$.
Similarly,the exterior angle at vertex $C$ is $\angle ACE = 80^{\circ}$. Thus,the interior angle $\angle ACB = 180^{\circ} - 80^{\circ} = 100^{\circ}$.
In $\Delta ABC$,the sum of all interior angles is $180^{\circ}$. Therefore,$\angle A + \angle ABC + \angle ACB = 180^{\circ}$.
Substituting the values,$\angle A + 40^{\circ} + 100^{\circ} = 180^{\circ}$.
$\angle A + 140^{\circ} = 180^{\circ}$.
$\angle A = 180^{\circ} - 140^{\circ} = 40^{\circ}$.

(A) Let the measure of the angle be $x^o$.
Since the sum of complementary angles is $90^o$,the measure of its complementary angle is $(90 - x)^o$.
According to the problem,the angle is $\frac{5}{4}$ times its complementary angle:
$x = \frac{5}{4}(90 - x)$
Multiply both sides by $4$:
$4x = 5(90 - x)$
$4x = 450 - 5x$
Add $5x$ to both sides:
$9x = 450$
Divide by $9$:
$x = 50$
Therefore,the measure of the angle is $50^o$.

(B) According to the Exterior Angle Theorem,the measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.
In $\Delta ABC$,$\angle ACD$ is the exterior angle at vertex $C$.
The interior opposite angles are $\angle A$ and $\angle B$.
Therefore,$\angle ACD = \angle A + \angle B$.
Given $\angle A = 70^{\circ}$ and $\angle B = 40^{\circ}$.
$\angle ACD = 70^{\circ} + 40^{\circ} = 110^{\circ}$.

(C) The sum of all interior angles of a triangle is always $180^{\circ}$.
In $\Delta PQR$,we have $\angle P + \angle Q + \angle R = 180^{\circ}$.
Given $\angle P = 42^{\circ}$ and $\angle Q = 75^{\circ}$.
Substituting these values: $42^{\circ} + 75^{\circ} + \angle R = 180^{\circ}$.
$117^{\circ} + \angle R = 180^{\circ}$.
$\angle R = 180^{\circ} - 117^{\circ} = 63^{\circ}$.

(A) Let the measure of the angle be $x^o$.
Two angles are supplementary if their sum is $180^o$.
Therefore,the supplementary angle of $x^o$ is $(180 - x)^o$.
According to the problem,the angle is one-eighth of its supplementary angle:
$x = \frac{1}{8}(180 - x)$
Multiply both sides by $8$:
$8x = 180 - x$
Add $x$ to both sides:
$9x = 180$
Divide by $9$:
$x = 20$
Thus,the measure of the angle is $20^o$.

(A) Two angles are said to be supplementary if the sum of their measures is $180^{\circ}$.
Let the supplementary angle of $52^{\circ}$ be $x$.
According to the definition,$52^{\circ} + x = 180^{\circ}$.
Therefore,$x = 180^{\circ} - 52^{\circ}$.
$x = 128^{\circ}$.
Thus,the measure of the supplementary angle is $128^{\circ}$.

(B) Two angles are said to be complementary if the sum of their measures is $90^{\circ}$.
Let the measure of the complementary angle be $x$.
According to the definition,$x + 44^{\circ} = 90^{\circ}$.
Subtracting $44^{\circ}$ from both sides,we get $x = 90^{\circ} - 44^{\circ}$.
Therefore,$x = 46^{\circ}$.

(C) Step $1$: Find the complementary angle of $62^{\circ}$.
The sum of two complementary angles is $90^{\circ}$.
Complementary angle = $90^{\circ} - 62^{\circ} = 28^{\circ}$.
Step $2$: Find the supplementary angle of the result obtained in Step $1$.
The sum of two supplementary angles is $180^{\circ}$.
Supplementary angle = $180^{\circ} - 28^{\circ} = 152^{\circ}$.

(D) Step $1$: Find the supplementary angle of $128^{\circ}$.
The sum of two supplementary angles is $180^{\circ}$.
Supplementary angle = $180^{\circ} - 128^{\circ} = 52^{\circ}$.
Step $2$: Find the complementary angle of the result obtained in Step $1$.
The sum of two complementary angles is $90^{\circ}$.
Complementary angle = $90^{\circ} - 52^{\circ} = 38^{\circ}$.

(A) Let the measure of the acute angle be $x^{\circ}$.
The supplementary angle of $x^{\circ}$ is $(180 - x)^{\circ}$.
The complementary angle of $x^{\circ}$ is $(90 - x)^{\circ}$.
The difference between the supplementary angle and the complementary angle is:
Difference $= (180 - x)^{\circ} - (90 - x)^{\circ}$
Difference $= 180 - x - 90 + x$
Difference $= 180 - 90 = 90^{\circ}$.