Класс 9 RD Sharma Solutions – Глава 8 Введение в линии и углы – Упражнение 8.4 | Набор 3
Вопрос 21. На данном рисунке трансверсаль l пересекает две прямые n, ∠4 = 110° и ∠7 = 65°. Изм||н?
Решение:
The figure is given as follows:
It is given that l is a transversal to lines m and n. Also,
∠4 = 110° and ∠7 = 65°
We need to check whether m || n or not.
We have ∠7 = 65°.
Also, ∠7 and ∠5 are vertically opposite angles, thus, these two must be equal. That is,
∠5 = 65° ………(i)
Also, ∠4 = 110°.
Adding this equation to (i), we get:
∠4 + ∠5 = 110° + 65°
∠4 + ∠5 = 175°
But these are the consecutive interior angles which are not supplementary.
Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.
Thus, m is not parallel to n.
Вопрос 22. Какие пары прямых на данном рисунке параллельны? Приведены причины.
Решение:
The figure is given as follows:
We have ∠BCD = 115° and ∠ADC = 65°.
Clearly,
∠BCD + ∠ADC = 115° + 65°
∠BCD + ∠ADC = 180°.
These are the pair of consecutive interior angles.
Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.
Thus, AD || BC.
Similarly, we have ∠DAB = 115° and ∠ADC = 65°.
Clearly,
∠DAB + ∠ADC = 115° + 65°
∠DAB + ∠ADC = 180°.
These are the pair of consecutive interior angles.
Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.
Thus, AB || CD.
Hence, the lines which are parallel are as follows:
AD || BC and AB || CD.
Вопрос 23. Если l, m, n — три прямые, такие что l||m и n⊥l. Докажите, что n⊥m.
Решение:
The figure can be drawn as follows:
Here, l || m and n ⊥ l
We need to prove that n ⊥ m.
It is given that n ⊥ l, therefore,
∠1 = 90° ……(i)
We have l || m, thus, ∠1 and ∠2 are the corresponding angles. Therefore, these must be equal. That is,
∠1 = ∠2
From equation (i), we get:
∠2 = 90°
Therefore, n ⊥ m.
Hence, proved.
Вопрос 24. На данном рисунке плечи BA и BC ∠ABC соответственно параллельны плечам ED и EF ∠DEF. Докажите, что ∠ABC= ∠DEF
Решение:
The figure is given as follows:
It is given that, arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF..
We need to show that ∠ABC = ∠DEF
Let us extend BC to meet EF.
We have AB || DE ∠ABC and ∠DEF are corresponding angles, these two should be equal.
Therefore,
∠ABC = ∠DEF
Hence, proved.
Вопрос 25. На данном рисунке плечи BA и BC дуги ∠ABC параллельны соответственно плечам ED и EF дуги ∠DEF. Докажите, что ∠ABC+ ∠DEF= 180°.
Решение:
The figure is given as follows:
It is given that, arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF.
We need to show that ∠ABC + ∠DEF = 180°
Let us extend BC to meet ED at point P.
We have AB || DE and BP || EF. So, ∠BPE and ∠PEF are corresponding angles, these two should be equal.
Therefore,
∠BPE = ∠PEF
Also, we have AB || PE. So, ∠ABP and ∠BPE are consecutive interior angles, these two must be supplementary.
Therefore,
∠ABP + ∠BPE = 180°
∠ABC + ∠PEF = 180°
∠ABC + ∠DEF = 180°
Hence, proved.
Вопрос 26. Какие из следующих утверждений верны (T), а какие ложны (F)? Назови причины.
(i) Если две прямые пересекаются секущей, то соответствующие углы равны.
(ii) Если две параллельные прямые пересекаются секущей, то параллельные внутренние углы равны.
(iii) Две прямые, перпендикулярные одной и той же прямой, перпендикулярны друг другу.
(iv) Две прямые, параллельные одной и той же прямой, параллельны друг другу.
(v) Если две параллельные прямые пересекаются секущей, то внутренние углы по одну сторону от этой секущей равны.
Решение:
(i) Statement: If two lines are intersected by a transversal, then corresponding angles are equal.
False
Reason: The above statement holds good if the lines are parallel only.
(ii) Statement: If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
True
Reason: Letlandmare two parallel lines.
And transversaltintersectslandmmakinga two pair of alternate interior angles, ∠1, ∠2 and ∠3, ∠4
We need to prove that ∠1 = ∠2 and ∠3 = ∠4.
We have,
∠2 = ∠5 (Vertically opposite angles)
And, ∠1 = ∠5 (corresponding angles)
Therefore,
∠1 = ∠2 (Vertically opposite angles)
Again, ∠3 = ∠6 (corresponding angles)
Hence, ∠1 = ∠2 and ∠3 = ∠4.
(iii) Statement: Two lines perpendicular to the same line are perpendicular to each other.
False
Reason:The figure can be drawn as follows:
Here, l ⊥ n and m ⊥ n
It is given that l ⊥ n, therefore,
∠1 = 90° …….(i)
Similarly, we have m ⊥n, therefore,
∠2 = 90° …….(ii)
From (i) and (ii), we get:
∠1 = ∠2
But these are the pair of corresponding angles.
Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.
Thus, we can say that l || m.
(iv) Statement: Two lines parallel to the same line are parallel to each other.
True
Reason: The figure is given as follows:
It is given that l || m and m || n
We need to show that l || m
We have l || m, thus, corresponding angles should be equal.
That is,
∠1 = ∠2
Similarly,
∠3 = ∠2
Therefore,
∠1 = ∠3
But these are the pair of corresponding angles.
Therefore, l || m.
(v) Statement: If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are equal.
False
Reason: Theorem states: If a transversal intersects two parallel lines then the pair of alternate interior angles is equal.
Вопрос 27. Заполните пропуски в каждом из следующих утверждений, чтобы утверждение было верным:
(i) Если две параллельные прямые пересекаются секущей, то каждая пара соответствующих углов равна …
(ii) Если две параллельные прямые пересекаются секущей, то внутренние углы по одну сторону от этой секущей равны ….
(iii) Две прямые, перпендикулярные одной и той же прямой, … друг к другу.
(iv) Две прямые, параллельные одной и той же прямой, … друг к другу.
(v) Если секущая пересекает пару прямых так, что пара противоположных углов равны, то прямые …
(vi) Если секущая пересекает пару прямых на таком расстоянии, что сумма внутренних углов по одну и ту же сторону от секущей равна 180°, то прямые …
Решение:
(i) If two parallel lines are intersected by a transversal, then corresponding angles are equal.
(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.
(iii) Two lines perpendicular to the same line are parallel to each other.
(iv) Two lines parallel to the same line are parallel to each other.
(v) If a transversal intersects a pair of lines in such a way that a pair of interior angles is equal, then the lines are parallel.
(vi) If a transversal intersects a pair of lines in such a way that a pair of interior angles on the same side of transversal is 180°, then the lines are parallel.