12 multiple-choice questions, progressively harder.
Distinct coplanar lines ℓ,m\ell,mℓ,m are cut by ttt at distinct points, as numbered in the diagram. Given m∠2=m∠6=44∘m\angle2=m\angle6=44^\circm∠2=m∠6=44∘, what follows?
Solution
Correct answer: B
Angles 2 and 6 are corresponding angles at the two crossings.
Their measures are equal, so their angles are congruent. The corresponding-angle converse gives ℓ∥m\ell\parallel mℓ∥m.
Distinct coplanar lines ℓ,m\ell,mℓ,m are cut by ttt at distinct points, as numbered in the diagram. Angles 3 and 5 both measure 112∘112^\circ112∘. Which reason establishes parallelism?
Correct answer: D
Angles 3 and 5 lie between the two lines and on opposite sides of the transversal.
They are alternate interior angles. Their congruence establishes ℓ∥m\ell\parallel mℓ∥m by the alternate interior converse.
Distinct coplanar lines ℓ,m\ell,mℓ,m are cut by ttt at distinct points, as numbered in the diagram. Angles 4 and 5 measure 66∘66^\circ66∘ and 114∘114^\circ114∘. Which statement is justified?
Correct answer: A
Angles 4 and 5 are same-side interior angles. Their measures satisfy
66∘+114∘=180∘.66^\circ+114^\circ=180^\circ.66∘+114∘=180∘.
The same-side interior converse establishes parallelism.
Distinct coplanar lines ℓ,m\ell,mℓ,m are cut by ttt at distinct points, as numbered in the diagram. Angles 1 and 8 measure 91∘91^\circ91∘ and 89∘89^\circ89∘. What is the applicable parallelism test?
Correct answer: C
Angles 1 and 8 are outside the two lines and on the same side of ttt.
91∘+89∘=180∘.91^\circ+89^\circ=180^\circ.91∘+89∘=180∘.
The same-side exterior converse gives ℓ∥m\ell\parallel mℓ∥m.
Distinct coplanar lines ℓ,m\ell,mℓ,m are cut by ttt at distinct points, as numbered in the diagram. Which equality alone does not test whether ℓ∥m\ell\parallel mℓ∥m?
Angles 1 and 3 are vertical angles at the upper crossing. Their equality is true regardless of the direction of mmm.
The other pairs are corresponding or alternate interior pairs at different crossings and do give converse tests.
Distinct lines ℓ,m,n\ell,m,nℓ,m,n lie in one plane. Both ℓ\ellℓ and mmm are perpendicular to nnn. What follows?
The distinct coplanar perpendiculars meet nnn at distinct points.
Corresponding angles at these crossings are both right, so the corresponding-angle converse gives ℓ∥m\ell\parallel mℓ∥m.
In one plane, ℓ∥m\ell\parallel mℓ∥m and transversal ttt is perpendicular to ℓ\ellℓ. What follows about ttt and mmm?
A corresponding angle at mmm equals the right angle at ℓ\ellℓ by the forward corresponding-angle theorem.
Thus their angle is 90∘90^\circ90∘, so t⊥mt\perp mt⊥m.
Which condition belongs in the rule that two distinct lines perpendicular to the same line are parallel?
The proof compares corresponding right angles in a plane.
Without coplanarity, different spatial directions can each meet the same line at a right angle, so the planar conclusion is not guaranteed.
In the diagram, H,KH,KH,K lie on mmm, PH⊥mPH\perp mPH⊥m, PH=14PH=14PH=14 cm, and PK=20PK=20PK=20 cm. What is the distance from PPP to mmm?
Point-to-line distance is defined as the length of the perpendicular segment.
PHPHPH is the stated perpendicular segment, so the distance is 141414 cm. The oblique segment PKPKPK is not used in that definition.
Two parallel lines have a specified perpendicular joining segment PQPQPQ of length 161616 cm. An oblique joining segment RSRSRS has length 191919 cm. Which given length measures perpendicular separation?
A perpendicular joining segment has one endpoint on each parallel and meets both at right angles.
The specified segment PQPQPQ has those properties, so its 161616 cm length measures the stated perpendicular separation.
An angle-copy construction through an external point produces congruent corresponding angles with the original line and one transversal. What justifies parallelism?
The accepted copying fact supplies a congruent corresponding pair at distinct crossings.
The corresponding-angle converse turns that angle fact into parallelism. The copy alone does not show parallelism; the converse supplies that step.
Distinct coplanar lines ℓ,m\ell,mℓ,m are cut by ttt at distinct points, as numbered in the diagram. Which additional given is sufficient to prove ℓ∥m\ell\parallel mℓ∥m?
Angles 2 and 7 are same-side exterior angles, so their supplementarity supplies a converse test.
The other facts concern one crossing only: a right-angle condition at one crossing, vertical angles, or a linear pair. They do not establish the second line direction.
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