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LINES AND ANGLES - Q&A

Exercise 6.1

1. In Fig. 6.13, lines AB and CD intersect at O. If ∠ AOC + ∠ BOE = 70° and ∠ BOD = 40°, find ∠ BOE and reflex ∠ COE.

Given:
Lines AB and CD intersect at O.
∠ AOC + ∠ BOE = 70° ... (i)
∠ BOD = 40°

Step 1: Find ∠ AOC.
Since lines AB and CD intersect at O, vertically opposite angles are equal.
∠ AOC = ∠ BOD
Therefore, ∠ AOC = 40°.

Step 2: Find ∠ BOE.
Substitute the value of ∠ AOC in equation (i):
40° + ∠ BOE = 70°
∠ BOE = 70° - 40°
∠ BOE = 30°.

Step 3: Find reflex ∠ COE.
We know that AOB is a straight line, so the sum of angles on it is 180°.
∠ AOC + ∠ COE + ∠ BOE = 180°
From (i), we know ∠ AOC + ∠ BOE = 70°.
So, 70° + ∠ COE = 180°
∠ COE = 180° - 70° = 110°.

Reflex ∠ COE = 360° - ∠ COE
Reflex ∠ COE = 360° - 110° = 250°.

Answer: ∠ BOE = 30° and Reflex ∠ COE = 250°.

2. In Fig. 6.14, lines XY and MN intersect at O. If ∠ POY = 90° and a : b = 2 : 3, find c.

Given:
Lines XY and MN intersect at O.
∠ POY = 90°
a : b = 2 : 3

Step 1: Find values of a and b.
Since XOY is a straight line, the sum of angles on it is 180°.
∠ POX + ∠ POY = 180°
∠ POX + 90° = 180°
∠ POX = 90°
From the figure, ∠ POX = a + b.
So, a + b = 90°.

Let a = 2x and b = 3x.
2x + 3x = 90°
5x = 90°
x = 90° / 5 = 18°.

So, a = 2(18°) = 36°
b = 3(18°) = 54°.

Step 2: Find c.
MN is a straight line, so the sum of angles on it is 180°.
b + c = 180° (Linear Pair)
54° + c = 180°
c = 180° - 54°
c = 126°.

Answer: c = 126°.

3. In Fig. 6.15, ∠ PQR = ∠ PRQ, then prove that ∠ PQS = ∠ PRT.

Given: ∠ PQR = ∠ PRQ
To Prove: ∠ PQS = ∠ PRT

Proof:
ST is a straight line.
∠ PQS + ∠ PQR = 180° (Linear Pair) ... (i)
∠ PRT + ∠ PRQ = 180° (Linear Pair) ... (ii)

From (i) and (ii), both sums are 180°, so we can equate them:
∠ PQS + ∠ PQR = ∠ PRT + ∠ PRQ

Since ∠ PQR = ∠ PRQ (Given), we can cancel them from both sides:
∠ PQS = ∠ PRT
Hence Proved.

4. In Fig. 6.16, if x + y = w + z, then prove that AOB is a line.

Given: x + y = w + z
To Prove: AOB is a line.

Proof:
The sum of all angles around a point is 360°.
So, x + y + w + z = 360°

Substitute (w + z) with (x + y) since they are equal:
(x + y) + (x + y) = 360°
2(x + y) = 360°
x + y = 180°

Since x and y are adjacent angles and their sum is 180°, they form a linear pair.
Therefore, AOB is a straight line.
Hence Proved.

5. In Fig. 6.17, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ ROS = 1/2 (∠ QOS - ∠ POS).

Given:
POQ is a straight line.
OR ⊥ PQ, so ∠ ROQ = 90° and ∠ ROP = 90°.

Proof:
From the figure, ∠ QOS = ∠ ROQ + ∠ ROS
Since ∠ ROQ = 90°, we have:
∠ QOS = 90° + ∠ ROS ... (i)

Also, ∠ POS = ∠ ROP - ∠ ROS
Since ∠ ROP = 90°, we have:
∠ POS = 90° - ∠ ROS ... (ii)

Subtract equation (ii) from equation (i):
∠ QOS - ∠ POS = (90° + ∠ ROS) - (90° - ∠ ROS)
∠ QOS - ∠ POS = 90° + ∠ ROS - 90° + ∠ ROS
∠ QOS - ∠ POS = 2∠ ROS

Dividing both sides by 2:
∠ ROS = 1/2 (∠ QOS - ∠ POS)
Hence Proved.

6. It is given that ∠ XYZ = 64° and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ ZYP, find ∠ XYQ and reflex ∠ QYP.

Step 1: Draw the figure.
Draw angle XYZ = 64°. Extend line XY to P. Draw ray YQ bisecting ∠ ZYP.

Step 2: Find ∠ ZYP.
Since XYP is a straight line:
∠ XYZ + ∠ ZYP = 180°
64° + ∠ ZYP = 180°
∠ ZYP = 180° - 64° = 116°.

Step 3: Find ∠ QYP and ∠ ZYQ.
Since YQ bisects ∠ ZYP:
∠ QYP = ∠ ZYQ = ∠ ZYP / 2
∠ QYP = 116° / 2 = 58°.

Step 4: Find ∠ XYQ.
∠ XYQ = ∠ XYZ + ∠ ZYQ
∠ XYQ = 64° + 58° = 122°.

Step 5: Find Reflex ∠ QYP.
Reflex ∠ QYP = 360° - ∠ QYP
Reflex ∠ QYP = 360° - 58° = 302°.

Answer: ∠ XYQ = 122° and Reflex ∠ QYP = 302°.

Exercise 6.2

1. In Fig. 6.28, find the values of x and y and then show that AB || CD.

Step 1: Find x.
From the figure, the angle 50° and x form a linear pair on the transversal.
50° + x = 180°
x = 180° - 50° = 130°.

Step 2: Find y.
y and 130° are vertically opposite angles.
y = 130°.

Step 3: Show AB || CD.
We found x = 130° and y = 130°.
Therefore, x = y.
These are alternate interior angles.
Since the alternate interior angles are equal, the lines AB and CD are parallel.
Hence, AB || CD.

2. In Fig. 6.29, if AB || CD, CD || EF and y : z = 3 : 7, find x.

Given:
AB || CD and CD || EF.
This implies AB || EF (Lines parallel to the same line are parallel to each other).
y : z = 3 : 7.

Step 1: Establish relationship between angles.
Since AB || EF, angles x and z are alternate interior angles.
So, x = z.

Since AB || CD, angles x and y are consecutive interior angles (same side of transversal).
x + y = 180°.

Step 2: Solve for z.
Substitute x = z in the equation x + y = 180°:
z + y = 180°.

We are given y : z = 3 : 7.
Let y = 3k and z = 7k.
3k + 7k = 180°
10k = 180°
k = 18°.

So, z = 7k = 7 × 18° = 126°.

Step 3: Find x.
Since x = z, then x = 126°.

Answer: x = 126°.

3. In Fig. 6.30, if AB || CD, EF ⊥ CD and ∠ GED = 126°, find ∠ AGE, ∠ GEF and ∠ FGE.

Given:
AB || CD
EF ⊥ CD, so ∠ FED = 90°
∠ GED = 126°

Step 1: Find ∠ AGE.
Since AB || CD, alternate interior angles are equal.
∠ AGE = ∠ GED
∠ AGE = 126°.

Step 2: Find ∠ GEF.
From the figure, ∠ GED = ∠ GEF + ∠ FED
126° = ∠ GEF + 90°
∠ GEF = 126° - 90°
∠ GEF = 36°.

Step 3: Find ∠ FGE.
G lies on the straight line AB.
∠ AGE + ∠ FGE = 180° (Linear Pair)
126° + ∠ FGE = 180°
∠ FGE = 180° - 126°
∠ FGE = 54°.

Answer: ∠ AGE = 126°, ∠ GEF = 36°, ∠ FGE = 54°.

4. In Fig. 6.31, if PQ || ST, ∠ PQR = 110° and ∠ RST = 130°, find ∠ QRS.

[Hint: Draw a line parallel to ST through point R.]

Construction: Draw a line RU through R such that RU || ST.
Since PQ || ST and RU || ST, then PQ || RU.

Step 1: Find angles related to parallel lines.
Since ST || RU, angles on the same side of transversal SR sum to 180°.
∠ RST + ∠ SRU = 180°
130° + ∠ SRU = 180°
∠ SRU = 50°.

Since PQ || RU, alternate interior angles are equal.
∠ PQR = ∠ QRU
∠ QRU = 110°.

Step 2: Find ∠ QRS.
From the figure, ∠ QRU = ∠ QRS + ∠ SRU
110° = ∠ QRS + 50°
∠ QRS = 110° - 50°
∠ QRS = 60°.

Answer: ∠ QRS = 60°.

5. In Fig. 6.32, if AB || CD, ∠ APQ = 50° and ∠ PRD = 127°, find x and y.

Given:
AB || CD
∠ APQ = 50°
∠ PRD = 127°

Step 1: Find x.
Since AB || CD, alternate interior angles are equal.
∠ APQ = ∠ PQR
50° = x
So, x = 50°.

Step 2: Find y.
Since AB || CD, alternate interior angles are equal.
∠ APR = ∠ PRD
From the figure, ∠ APR = ∠ APQ + y = 50° + y.
So, 50° + y = 127°
y = 127° - 50°
y = 77°.

Answer: x = 50° and y = 77°.

6. In Fig. 6.33, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.

Given: Mirror PQ || Mirror RS.
To Prove: AB || CD.

Construction:
Draw normal BM ⊥ PQ at B.
Draw normal CN ⊥ RS at C.

Proof:
1. Since PQ || RS and BM and CN are perpendiculars to parallel lines, BM || CN.

2. Laws of reflection: Angle of incidence = Angle of reflection.
Let angles at B be ∠ 1 and ∠ 2 (where 1 is incidence, 2 is reflection). So, ∠ 1 = ∠ 2.
Let angles at C be ∠ 3 and ∠ 4 (where 3 is incidence, 4 is reflection). So, ∠ 3 = ∠ 4.

3. Since BM || CN and BC is a transversal:
∠ 2 = ∠ 3 (Alternate Interior Angles).

4. From the above equalities:
Since ∠ 1 = ∠ 2 and ∠ 3 = ∠ 4 and ∠ 2 = ∠ 3,
Then ∠ 1 = ∠ 2 = ∠ 3 = ∠ 4.

5. Consider the whole angles:
∠ ABC = ∠ 1 + ∠ 2 = 2(∠ 2)
∠ BCD = ∠ 3 + ∠ 4 = 2(∠ 3)
Since ∠ 2 = ∠ 3, then 2(∠ 2) = 2(∠ 3).
Therefore, ∠ ABC = ∠ BCD.

6. ∠ ABC and ∠ BCD are alternate interior angles formed by transversal BC intersecting lines AB and CD.
Since alternate interior angles are equal, AB || CD.
Hence Proved.

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Quick Review Flashcards - Click to flip and test your knowledge!

Question

Term: Line-segment

Answer

Definition: A part or portion of a line with two distinct end points.

Question

What is the geometric definition of a 'ray'?

Answer

A part of a line that has exactly one end point.

Question

If three or more points lie on the same line, what are they called?

Answer

Collinear points.

Question

Points that do not lie on the same line are categorized as _____.

Answer

non-collinear points

Question

How is an angle formed in geometry?

Answer

An angle is formed when two rays originate from the same end point.

Question

Term: Arms of the angle

Answer

Definition: The two rays that originate from a common end point to form an angle.

Question

What is the 'vertex' of an angle?

Answer

The common end point from which the two rays of an angle originate.

Question

What is the measurement range for an acute angle $x$?

Answer

$0^\circ < x < 90^\circ$

Question

What is the exact measurement of a right angle?

Answer

$90^\circ$

Question

An angle $z$ is classified as obtuse if its measure falls within which range?

Answer

$90^\circ < z < 180^\circ$

Question

What is the exact measurement of a straight angle?

Answer

$180^\circ$

Question

An angle $t$ is defined as a reflex angle if it satisfies which inequality?

Answer

$180^\circ < t < 360^\circ$

Question

Two angles are called complementary if the sum of their measures equals _____.

Answer

$90^\circ$

Question

Two angles are called supplementary if the sum of their measures equals _____.

Answer

$180^\circ$

Question

What three conditions must be met for two angles to be considered 'adjacent'?

Answer

They must have a common vertex, a common arm, and non-common arms on different sides of the common arm.

Question

If $\angle ABD$ and $\angle DBC$ are adjacent angles sharing the arm $BD$, what is the measure of the resulting angle $\angle ABC$?

Answer

$\angle ABC = \angle ABD + \angle DBC$

Question

What term describes adjacent angles whose non-common arms form a single straight line?

Answer

Linear pair of angles.

Question

How are vertically opposite angles formed?

Answer

They are formed when two lines intersect each other at a single point.

Question

What defines 'parallel lines' in a plane?

Answer

Lines that never intersect, even when extended indefinitely in both directions.

Question

What is the 'distance between two parallel lines'?

Answer

The constant length of the common perpendiculars at any point between the lines.

Question

Axiom 6.1: If a ray stands on a line, then the sum of the two adjacent angles so formed is _____.

Answer

$180^\circ$

Question

Axiom 6.2: If the sum of two adjacent angles is $180^\circ$, what can be concluded about their non-common arms?

Answer

The non-common arms form a line.

Question

The combination of Axiom 6.1 and Axiom 6.2 is collectively known as the _____.

Answer

Linear Pair Axiom

Question

Theorem 6.1: If two lines intersect each other, then the _____ angles are equal.

Answer

vertically opposite

Question

In the proof of Theorem 6.1, which axiom is used to state that $\angle AOC + \angle AOD = 180^\circ$ when OA stands on line CD?

Answer

Linear pair axiom.

Question

According to Theorem 6.6, if line $m \parallel$ line $l$ and line $n \parallel$ line $l$, what is the relationship between $m$ and $n$?

Answer

Line $m$ is parallel to line $n$.

Question

How does an angle bisector affect an angle?

Answer

It divides the angle into two equal parts.

Question

In Example 2, if ray $OR$ bisects $\angle POS$ (where $\angle POS = x$), what is the measure of $\angle ROS$?

Answer

$\frac{x}{2}$

Question

In Example 2, if $OR$ and $OT$ are bisectors of a linear pair of angles, what is the resulting measure of $\angle ROT$?

Answer

$90^\circ$

Question

Based on Example 3, what is the sum of all angles formed by rays originating from a single point (full rotation)?

Answer

$360^\circ$

Question

If transversal $AD$ intersects lines $PQ$ and $RS$ such that corresponding angles are equal, what is the relationship between $PQ$ and $RS$?

Answer

$PQ \parallel RS$ (Converse of corresponding angles axiom).

Question

When a transversal intersects two parallel lines, what is the sum of the interior angles on the same side of the transversal?

Answer

$180^\circ$

Question

If $AB \parallel CD$ and $CD \parallel EF$, what is the geometric relationship between $AB$ and $EF$?

Answer

$AB \parallel EF$

Question

In Figure 6.11, the sum of $\angle POQ$, $\angle QOR$, $\angle SOR$, and $\angle POS$ is _____.

Answer

$360^\circ$

Question

If lines $PQ$ and $RS$ intersect at $O$ and $\angle POR : \angle ROQ = 5:7$, what is the measure of $\angle POR$?

Answer

$75^\circ$

Question

In Figure 6.13, if lines $AB$ and $CD$ intersect at $O$ and $\angle BOD = 40^\circ$, what is the measure of its vertically opposite angle $\angle AOC$?

Answer

$40^\circ$

Question

What determines if two angles are a 'linear pair' rather than just 'adjacent'?

Answer

The two angles must be adjacent and their non-common arms must form a straight line.

Question

Concept: Vertically Opposite Angles

Answer

Property: When two lines intersect, the angles opposite the vertex are always equal.

Question

If the sum of two angles is $180^\circ$, they are called _____ angles.

Answer

supplementary

Question

What is the result of adding two adjacent angles if their non-common arms are opposite rays?

Answer

$180^\circ$

Question

When solving for angles between parallel lines, 'interior angles on the same side of the transversal' are also called _____ angles.

Answer

consecutive interior (or co-interior)

Question

In geometry, what does the symbol $\perp$ represent?

Answer

Perpendicularity (intersecting at a $90^\circ$ angle).

Question

If ray $YQ$ bisects $\angle ZYP$ and $\angle ZYP = 116^\circ$, what is the measure of $\angle QYP$?

Answer

$58^\circ$

Question

Theorem 6.6 establishes that lines parallel to the same line are parallel to _____.

Answer

each other

Question

In Example 4, if $PQ \parallel RS$ and a line $AB$ is drawn parallel to $PQ$ through $M$, why is $AB$ also parallel to $RS$?

Answer

Because lines parallel to the same line are parallel to each other.

Question

How are 'alternate interior angles' related when a transversal intersects two parallel lines?

Answer

They are equal.

Question

If $x+y = 180^\circ$ for two adjacent angles, then the non-common arms of these angles lie on the same _____.

Answer

line

Question

What is the reflex angle of an angle measuring $60^\circ$?

Answer

$300^\circ$

Question

If $\angle POY = 90^\circ$ and angles $a$ and $b$ are adjacent such that $a+b = 90^\circ$, what is the relationship between $a$ and $b$?

Answer

They are complementary angles.

Question

When two lines intersect, how many pairs of vertically opposite angles are formed?

Answer

Two pairs.

Question

What logic is used to prove $\angle AOC = \angle BOD$ in Theorem 6.1 after establishing both sum to $180^\circ$ with $\angle AOD$?

Answer

Euclid's Axiom: If equals are subtracted from equals, the remainders are equal.

Question

In Example 5, what is the given condition regarding rays $BE$ and $CG$?

Answer

They are parallel bisectors of corresponding angles $\angle ABQ$ and $\angle BCS$.