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MATHEMATICS [5 marks sum]

Triangles · ICSE STD 9 (ICSE - English Medium). 17 questions.

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[5 marks sum]

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17 questions · timed · auto-graded

Question 15 Marks
In the adjoining figure, $AC > AB$ and AD is the bisector of $\angle A$. Show that: $\angle ADC >\angle ADB$.
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Answer
Hint :$A C>A B \Rightarrow \angle B>\angle C$
$\Rightarrow \angle B+\angle B A D>\angle C+\angle C A D$
$\Rightarrow \angle A D C>\angle A D B$.
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Question 35 Marks
In the given figure, $AD = AB$ and AE bisects $\angle A$.
Prove that :
(i) $BE = ED$
(ii) $\angle ABD >\angle BCA$.
Image
Answer
[Hint. Prove that $\triangle A E B \cong \triangle A E D \Rightarrow \angle A B D=\angle A D B>\angle B C A$.]
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Question 45 Marks
If O is any point inside $\triangle ABC$, prove that $\angle BOC >\angle A$.
Answer
[Hint. Join AO and produce it to meet BC at D. Then, Ext. $\angle B O D>\angle B A O$ and Ext. $\angle C O D>\angle C A O$. Now, add.]
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Question 55 Marks
In the adjoining quadrilateral $ABCD , AB$ is the longest side and DC is the shortest side. Prove that :
(i) $\angle C >\angle A$
(ii) $\angle D >\angle B$.
Image
Answer
[Hint. Join AC. Then
$\begin{array}{l}
\text { In } \triangle A B C, A B>B C \Rightarrow \angle B C A>\angle C A B \\
\text { In } \triangle A C D, A D>D C \Rightarrow \angle D C A>\angle D A C\end{array}$
Adding (1) and (2), we get : $\angle C>\angle A$.
Similarly, $\angle D>\angle B$.]
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Question 65 Marks
In the adjoining figure, $PL \perp QR ; LQ = LS$ and $LR > LQ$. Show that $PR > PQ$.
Image
Answer
[Hint.
$\begin{array}{l}\triangle P L Q \equiv \triangle P L S \Rightarrow \angle 1=\angle 3 .
\text { But }, \angle 3>\angle 2 .\\\therefore \angle 1>\angle 2 .]\end{array}$
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Question 75 Marks
In the adjoining figure, in $\triangle A B C, O$ is any point in its interior. Show that: $OB + OC < AB + AC$.
Image
Answer
[Hint. Produce BO to meet AC in D.
$\begin{array}{l}\text { Now, } A B+A D>B D \Rightarrow A B+A D>O B+O D \\\text { In } \triangle C O D, \quad O D+D C>O C
\\\therefore A B+O D+(A D+D C)>O B+O C+O D \\\Rightarrow A B+A C>O B+O C .1\end{array}$
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Question 85 Marks
In the given figure, side $C A$ of $\triangle A B C$ has been produced to E . If $AC = AD = BD ; \angle ACD =46^{\circ}$ and $\angle BAE =x^{\circ}$; find the value of $x$.
Image
Answer
$x=69$
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Question 95 Marks
In the given figure, side $B A$ of $\triangle A B C$ has been produced to D such that $CD = CA$ and side CB has been produced to E . If $\angle BAC =106^{\circ}$ and $\angle ABE =128^{\circ}$, find $\angle BCD$.
Image
Answer
$\angle BCD =54^{\circ}$
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Question 105 Marks
In the given figure, $CA = CD = BD ; \angle DBC =35^{\circ}$ and $\angle DCA =x^{\circ}$. Find the value of $x$.
Image
Answer
$x=40$
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Question 125 Marks
In the given figure, $AB = AD ; CB = CD ; \angle A =42^{\circ}$ and $\angle C =108^{\circ}$, find $\angle ABC$.
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Answer
$105^{\circ}$
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Question 135 Marks
In the given figure, $AB = AC$. If BO and CO , the bisectors of $\angle B$ and $\angle C$ respectively meet at O and BC is produced to D , prove that $\angle BOC =\angle ACD$.
Image
Answer
self
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Question 145 Marks
In the given figure, $A B=A C$; $D$ is the mid-point of $B C$; $DP \perp BA$ and $DQ \perp CA$.
Prove that:
(i) $DP = DQ$
(ii) $AP = AQ$
(iii) AD bisects $\angle A$.
Image
Answer
self
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Question 155 Marks
In the given figure, $AB = AC ; \angle A =50^{\circ}$ and $\angle ACD =15^{\circ}$. Show that $B C=C D$.
Image
Answer
self
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Question 165 Marks
Ms Anu Gupta teaches mathematics to class 9 in a school. One day she drew a figure on the board in the class. She provided the following clues to the students. - $AB \| CD$ - O is the mid-points of AD
Image
Based on the above information, answer the following questions :
Q.1. $\triangle OAB \equiv \triangle ODC$ by which of the following congruent condition?
(a) SAS (b) ASA (c) SSS (d) RHS
Q.2. $\angle AOB =\angle DOC$ holds because:
(a) Alternate angles are equal (b) Corresponding angles are equal (c) Vertically opposite angles are equal (d) None of these
Q.3. Which of the following is correct?
(a) $\angle A =\angle C$ (b) $\angle B =\angle D$ (c) $\angle B =\angle C$ (d) $\angle AOB =\angle OCB$
Q.4. Which of the following is correct?
(a) $AO = OB$ (b) $AB = OB$ (c) $OD = CD$ (d) $OC = OB$
Q.5. Which of the following is not a congruent condition?
(a) ASA (b) SSS (c) AAA (d) AAS
Answer
1. (b) 2. (c) 3. (c) 4. (d) 5. (c)
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Question 175 Marks
Image
Case Study : Ms Anu Gupta teaches mathematics to class 9 in a school. One day she drew a figure on the board in the class. She provided the following clues to the students.
- $AB |\mid CD$
- $O$ is the mid-points of $A D$
Based on the above information, answer the following questions :
1. $\triangle OAB \cong \triangle ODC$ by which of the following congruent condition?
(a) SAS (b) ASA
(c) SSS (d) RHS
2. $\angle AOB =\angle DOC$ holds because:
(a) Alternate angles are equal (b) Corresponding angles are equal
(c) Vertically opposite angles are equal (d) None of these
3. Which of the following is correct?
(a) $\angle A =\angle C$ (b) $\angle B =\angle D$
(c) $\angle B =\angle C$ (d) $\angle AOB =\angle OCB$
4. Which of the following is correct?
(a) $AO = OB$ (b) $AB = OB$
(c) $OD = CD$ (d) $OC = OB$
5. Which of the following is not a congruent condition?
(a) ASA (b) SSS
(c) AAA (d) AAS
Answer
(1. b), (2. c), (3. c), (4. d), (5. c)
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[5 marks sum] - MATHEMATICS STD 9 Questions - Vidyadip