Parallel and Intersecting Lines | FIO

Question 12

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.

Question diagram 1
Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • Given that AB∥CDAB \parallel CD and CD∥EFCD \parallel EF, which implies all three lines are parallel to each other (AB∥CD∥EFAB \parallel CD \parallel EF).
  • Consecutive interior angles on the same side of a transversal are supplementary (sum to 180∘180^\circ). Using the transversal intersecting CDCD and EFEF, we can find yy.
  • Corresponding angles are equal for parallel lines. Using the transversal intersecting ABAB and CDCD, we can find xx from yy.

Step 1 · Find the value of yy

Given CD∥EFCD \parallel EF with transversal DEDE.Diagram 1

Since consecutive interior angles on the same side of the transversal sum to 180∘180^\circ:

y∘+55∘=180∘y∘=180∘−55∘y=125∘\begin{aligned} y^\circ + 55^\circ &= 180^\circ \\[0.6em] y^\circ &= 180^\circ - 55^\circ \\[0.6em] y &= 125^\circ \end{aligned}

Step 2 · Find the value of xx

Given AB∥CDAB \parallel CD with transversal BDBD.

Since x∘x^\circ and y∘y^\circ are corresponding angles:

x∘=y∘x=125∘\begin{aligned} x^\circ &= y^\circ \\[0.6em] x &= 125^\circ \end{aligned}
Answer

x=125∘,y=125∘x = 125^\circ, \quad y = 125^\circ

Common Mistakes
  • Interior Angle Relation: Incorrectly assuming consecutive interior angles on the same side of the transversal are equal instead of supplementary (y+55∘=180∘y + 55^\circ = 180^\circ).
  • Angle Pair Identification: Confusing corresponding angles (which are equal: x=yx = y) with alternate interior or co-interior angles.

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.]

← Back to Parallel and Intersecting Lines