Parallel and Intersecting Lines | FIO

Question 11

In Fig. 5.33, ∠ABC=45∘\angle ABC = 45^\circ and ∠IKJ=78∘\angle IKJ = 78^\circ. Find angles ∠GEH\angle GEH, ∠HEF\angle HEF, ∠FED\angle FED

Question diagram 1
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Solution
Understand the Question
  • We are given parallel lines intersected by two transversal lines, forming several angle relationships.
  • Vertically opposite angles are equal at any intersection of two straight lines.
  • Corresponding angles are equal when two parallel lines are cut by a transversal.
  • The sum of all angles forming a straight line at a single point is 180∘180^\circ.
  • By finding ∠GEH\angle GEH and ∠FED\angle FED using angle relationships with parallel lines, we can determine the remaining angle ∠HEF\angle HEF on the straight line GDGD.

Step 1 · Find ∠GEH\angle GEH and ∠FED\angle FED Using Corresponding Angles

Given lines IA∥GDIA \parallel GD.Diagram 1

For transversal HCHC: ∠KBE=∠ABC=45∘(Vertically opposite angles)\angle KBE = \angle ABC = 45^\circ \quad \text{(Vertically opposite angles)}

Since IA∥GDIA \parallel GD, corresponding angles are equal:

∠GEH=∠KBE=45∘\begin{aligned} \angle GEH &= \angle KBE \\ &= 45^\circ \end{aligned}

For transversal JFJF: ∠BKE=∠IKJ=78∘(Vertically opposite angles)\angle BKE = \angle IKJ = 78^\circ \quad \text{(Vertically opposite angles)}

Since IA∥GDIA \parallel GD, corresponding angles are equal:

∠FED=∠BKE=78∘\begin{aligned} \angle FED &= \angle BKE \\ &= 78^\circ \end{aligned}

Step 2 · Find ∠HEF\angle HEF Using Straight Line Angle Sum

Angles ∠GEH\angle GEH, ∠HEF\angle HEF, and ∠FED\angle FED lie on the straight line GDGD.

Sum of angles on a straight line is 180∘180^\circ: 45∘+78∘=123∘45^\circ + 78^\circ = 123^\circ

∠HEF=180∘−(45∘+78∘)=180∘−123∘=57∘\begin{aligned} \angle HEF &= 180^\circ - (45^\circ + 78^\circ) \\ &= 180^\circ - 123^\circ \\ &= 57^\circ \end{aligned}
Answer

∠GEH=45∘\angle GEH = 45^\circ, ∠HEF=57∘\angle HEF = 57^\circ, and ∠FED=78∘\angle FED = 78^\circ

Common Mistakes
  • Misidentifying Angle Pairs: Confusing corresponding angles with alternate interior angles when identifying relations across parallel lines.
  • Straight Line Sum Error: Forgetting that all three adjacent angles forming the straight line GDGD at point EE must add up to 180∘180^\circ.

More questions in FIO

Q1

List all the linear pairs and vertically opposite angles you observe in Fig. 5.3:

Q2

Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.

Q3

In Fig. 5.11, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.

(a) How did you spot the perpendicular lines?

(b) How did you spot the parallel lines?

Q4

In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.

Q5

Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.

(a) Did you find it challenging to draw some of them? (b) Which ones? (c) How did you do it?

Q6

In Fig. 5.13, which line is parallel to line aa — line bb or line cc? How do you decide this?

Q7

Can you draw a line parallel to ll, that goes through point AA? How will you do it with the tools from your geometry box? Describe your method.

Q8

Find the angles marked below.

Q9

Find the angle represented by aa.

Q10

In the figures below, what angles do xx and yy stand for?

Q11

In Fig. 5.33, ∠ABC=45∘\angle ABC = 45^\circ and ∠IKJ=78∘\angle IKJ = 78^\circ. Find angles ∠GEH\angle GEH, ∠HEF\angle HEF, ∠FED\angle FED

Q12

In Fig. 5.34, ABAB is parallel to CDCD and CDCD is parallel to EFEF. Also, EAEA is perpendicular to ABAB. If ∠BEF=55∘\angle BEF = 55^\circ, find the values of xx and yy.

Q13

What is the measure of angle ∠NOP\angle \text{NOP} in Fig. 5.35?

[Hint: Draw lines parallel to LM and PQ through points N and O.]

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