Fundamentals of Geometry

Question 1: Determine the value of “a” in the figure considering that the two horizontal lines are parallel to each other.

The angles ∠92° and ∠(3a + 5)° are alternate interior angles and are therefore equal.
So, ∠(3a + 5)° = ∠92°
or, 3a + 5 = 92
or, 3a = 87
or, a = 29

Question 2: In the given figure AB || DF and BC || EG. If ∠GEF = 64°, ∠ABH =  27° the find the value of ∠BEG

AB || DF and EH is transverse, so to it ∠ABH = ∠DEB 27° since they are corresponding angles.
Since ∠DEF is a straight angle
∠BEG = ∠DEF – ∠DEH – ∠GEF
or, ∠BEG = 180° – 64° – 27° = 89°

Question 3: In the given figure lines AB and CD intersect at O. OE is another straight line originating from the intersection point O such that ∠BOE = 80°. If ∠BOD = 30° then find the value of ∠COE.

Given ∠BOE = 80° and ∠BOD = 30°
So ∠DOE = 80° – 30° = 50°
In straight angle ∠AOB
∠AOE = ∠AOB – ∠BOE
∠AOE  = 180° – 80°  = 100°
Again ∠AOC = ∠BOD = 30° (Vertically opposite angles)
So ∠COE = ∠AOC + ∠AOE
or, ∠COE = 30° + 100° = 130°

Question 4: In the given figure AB || DE find the value of x.

Draw a line CF parallel to AB and DE.
AB || CF and BC is the transverse line
again DE || CF and CD is the transverse line
∠ABC and ∠FCB are supplementary angles
∠FCB = 180° – 120°  = 60°
again ∠CDE and ∠DCF are supplementary angles
∠DCF = 180° – 140°  = 40°
So x = ∠BCF + ∠DCF = 40° + 60° = 100°

Question 5: In the figure below PQ || RS, ∠BPQ = 40°, ∠BPR = 155° and ∠CRS = 70°. Find the value of ∠PAR.

∠APR = ∠APB – ∠BPR
∠APR = 180° – 155° = 25°
Again
∠RPQ = ∠BPR – ∠BPQ
∠RPQ = 115° – 40° = 115°
Given PQ || RS and PR is transverse
So ∠RPQ and ∠PRS are supplementary
∠PRS = 180° – 115° = 65°
∠ARP = ∠ARC – ∠CRS – ∠PRS
∠ARP = 180° – 70° – 65° = 45°
∠PAR = 180° – 25° – 45° = 110°

Question 6: In the given figure if BE || DF, the calculate the value of x and y.

We construct a line CG parallel to BE and DF
∠c = 360° – ∠CDF
∠c = 360° – 220° = 140°
CG || DF and CD is the transversal
∠b + ∠c = 180° (supplementary angles)
So, ∠b = 180° –  140° = 40°
Again BD || CG and CE is tthe transversal
∠a = ∠BEC = 40° (Alternate angles)
So, y = ∠a + ∠b = 40° + 40° = 80°
∠BCE = ∠BCD – ∠y
∠BCE = 180° – 80° = 100°
∠x = ∠BEC + ∠BCE
∠x = 100° + 40° = 140°

Question 7: Find the value of x in the given figure.

∠JKL = ∠JKD – ∠LKD
∠JKL = 180° – 78° = 102°
∠JLK = 180° – ∠JKL – ∠LJK
∠JLK = 180° – 53° – 102° =  45°
x = ∠JLK = 45° (vertically opposite angles)

Question 8: Find the values of x and y in the given figure.

Question 9: Find the values of x and y in the given figure.

∠KMD = ∠GKB = (3x – 5) congruent angles
∠GKJ = 180° – (3x – 5)
Since ∠GKJ = ∠EJA congruent angles
we have
180° – (3x – 5) = 2x
or, 5x =  185°
or, x = 37°

Question 10: Find the value of y from the given figure where AC || DF.

Since AC || DF and GH is the transversal
∠ABE = ∠BEF corresponding angles
So, 3x = 2x + 25
or, x = 25°
∠BEF = 2 x 25° + 25° = 75°
Now, ∠BEF = ∠DEG vertically opposite angles
So (y + 15)° = 75°
or, y = 60°

Question 11: For what value of Y is the seven-digit number 46393Y8 divisible by 11?

Construct FE parallel to AB and CD
Since FE || DC and CE is transversal
∠DCE = ∠FEC are supplementary angles
So ∠FEC = 180° – 135° = 45°
∠FEB = 360° – ∠CEB – ∠FEC
∠FEB = 360° – 220° – 45° = 95°
Since FE || AB and BE is transversal
∠FEB and ∠ABE are supplementary angles
So ∠ABE = 180° – 95° = 85°

Question 12: Find the value of x in the given figure.

JK || LM and AD is transversal
∠ABF = ∠ACL = 105° consecutive angles
∠GCD = 180° – 105° = 75° supplementary angles
∠CGD = ∠MGL – ∠DGL
∠CGD = 180° – 120° = 60°
In triangle CGD
∠CDG = 180° – 75° – 60° = 45°

1. Confusing adjacent angles with vertically opposite angles: Adjacent angles share a common arm, whereas vertically opposite angles are equal but do not share a common arm.
2. Assuming all pairs of angles add up to 180°: Only supplementary angles and certain angle pairs formed by intersecting or parallel lines sum to 180°.
3. Treating complementary angles as supplementary angles: Complementary angles add up to 90°, whereas supplementary angles add up to 180°.
4. Assuming vertically opposite angles are supplementary: Vertically opposite angles are always equal, not necessarily supplementary.
5. Using the wrong angle relationship in parallel lines: Identify whether the angles are corresponding, alternate interior, alternate exterior or co-interior before applying the correct property.
6. Assuming corresponding angles are supplementary: Corresponding angles are equal when the lines are parallel.
7. Assuming alternate angles are supplementary: Alternate interior and alternate exterior angles are equal when the lines are parallel.
8. Forgetting that co-interior angles are supplementary: Co-interior angles on the same side of the transversal always add up to 180° when the lines are parallel.
9. Applying parallel-line properties without confirming parallelism: Angle relationships such as corresponding and alternate angles hold only when the lines are parallel.
10. Ignoring the given angle relationships while forming equations: First identify all equal or supplementary angles before writing equations to find the unknown angle.

Question 1: Find the value of x in the given figure

Question 2: Find the value of x in the given figure

Question 3: Find the value of ∠BCD in the figure given.

Polygons

Quadrilaterals

Average

Introduction to Statistics