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LearnQuantitative AptitudeGeometry & Mensuration

04.01 · Geometry & Mensuration

Lines & Angles

Open lesson11 formulas10 questions

On the syllabus of

Formulas

At a glance 11 formulas

  1. 1Line and pointon a line: 180∘; round a point: 360∘\text{on a line: } 180^\circ;\ \text{round a point: } 360^\circvertically opposite angles are equal\text{vertically opposite angles are equal}
  2. 2Complement and supplementcomplement of x=90∘−x\text{complement of } x = 90^\circ - xsupplement of x=180∘−x\text{supplement of } x = 180^\circ - x
  3. 3Parallel linesA transversal across parallel lines makes corresponding angles equal and alternate angles equal; co-interior angles (same side, between the lines) add to 180∘180^\circ.
  4. 4Bend between parallels∠BED=∠ABE+∠CDE\angle BED = \angle ABE + \angle CDEAB∥CD; E between, on the A,C sideAB \parallel CD;\ E \text{ between, on the } A, C \text{ side}
  5. 5Angle sumA+B+C=180∘A + B + C = 180^\circ
  6. 6Exterior angleexterior angle at C=A+B\text{exterior angle at } C = A + B
  7. 7Isosceles triangleEqual sides face equal angles, so with apex angle AA each base angle is 90∘−A290^\circ - \frac{A}{2}. Chain these through shared sides.
  8. 8Co-interior bisectorsThe bisectors of two co-interior angles meet at 90∘90^\circ, and so do the bisectors of a linear pair.
  9. 9Bisectors meet at I∠BIC=90∘+A2\angle BIC = 90^\circ + \tfrac{A}{2}both external: 90∘−A2; one of each: A2\text{both external: } 90^\circ - \tfrac{A}{2};\ \text{one of each: } \tfrac{A}{2}
  10. 10Altitude and bisectorangle between them=∣B−C∣2\text{angle between them} = \dfrac{|B - C|}{2}both drawn from A to BC\text{both drawn from } A \text{ to } BC
  11. 11Five-point starthe five tip angles add to 180∘\text{the five tip angles add to } 180^\circ

Test yourself

10 questions · 3 easy · 4 medium · 3 hard

Tap an option to answer. The first tap counts.

  1. Q1Easy

    Lines PQ and RS are parallel. A transversal crosses PQ at X and RS at Y, and P and R lie on the same side of the transversal. If ∠PXY=64∘\angle PXY = 64^\circ, what is ∠RYX\angle RYX?

    Correct answer116∘116^\circ

    Why

    Co-interior angles: ∠PXY\angle PXY and ∠RYX\angle RYX sit between the parallels on the same side, so they add to 180∘180^\circ.

    180−64=116180 - 64 = 116; treating them as alternate angles gives the trap 64∘64^\circ.

    So ∠RYX=116∘\angle RYX = 116^\circ.

  2. Q2Easy

    In triangle ABC, side AC is produced beyond C, and the angle marked at C is the exterior angle between CB and the produced side. If angle A measures 50∘50^\circ and angle B measures 25∘25^\circ, what is the measure of the marked angle at C?

    Correct answer75∘75^\circ

    Why

    Exterior angle rule: the marked angle at C is angle A plus angle B, the two angles away from C.

    50+25=7550 + 25 = 75, with no detour through the inside angle of 105∘105^\circ.

    So the marked angle measures 75∘75^\circ.

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