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Triangles EXERCISE 7.1 Solutions
- Q1: In quadrilateral ACBD, $AC = AD$ and $AB$ bisects $\angle A$ (see Fig. 7.16). Show that $\triangle ABC \cong \triangle ABD$. What can you say about $BC$ and $BD$?
- Q2(i): ABCD is a quadrilateral in which $AD = BC$ and $\angle DAB = \angle CBA$ (see Fig. 7.17). Prove that (i) $\triangle ABD \cong \triangle BAC$
- Q2(ii): ABCD is a quadrilateral in which $AD = BC$ and $\angle DAB = \angle CBA$ (see Fig. 7.17). Prove that (ii) $BD = AC$
- Q2(iii): ABCD is a quadrilateral in which $AD = BC$ and $\angle DAB = \angle CBA$ (see Fig. 7.17). Prove that (iii) $\angle ABD = \angle BAC$.
- Q3: $AD$ and $BC$ are equal perpendiculars to a line segment $AB$ (see Fig. 7.18). Show that $CD$ bisects $AB$.
- Q4: $l$ and $m$ are two parallel lines intersected by another pair of parallel lines $p$ and $q$ (see Fig. 7.19). Show that $\triangle ABC \cong \triangle CDA$.
- Q5(i): Line $l$ is the bisector of an angle $\angle A$ and $B$ is any point on $l$. $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\angle A$ (see Fig. 7.20). Show that: (i) $\triangle APB \cong \triangle AQB$
- Q5(ii): Line $l$ is the bisector of an angle $\angle A$ and $B$ is any point on $l$. $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\angle A$ (see Fig. 7.20). Show that: (ii) $BP = BQ$ or $B$ is equidistant from the arms of $\angle A$.
- Q6: In Fig. 7.21, $AC = AE$, $AB = AD$ and $\angle BAD = \angle EAC$. Show that $BC = DE$.
- Q7(i): $AB$ is a line segment and $P$ is its mid-point. $D$ and $E$ are points on the same side of $AB$ such that $\angle BAD = \angle ABE$ and $\angle EPA = \angle DPB$ (see Fig. 7.22). Show that (i) $\triangle DAP \cong \triangle EBP$
- Q7(ii): $AB$ is a line segment and $P$ is its mid-point. $D$ and $E$ are points on the same side of $AB$ such that $\angle BAD = \angle ABE$ and $\angle EPA = \angle DPB$ (see Fig. 7.22). Show that (ii) $AD = BE$
- Q8(i): In right triangle ABC, right angled at $C$, $M$ is the mid-point of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM = CM$. Point $D$ is joined to point $B$ (see Fig. 7.23). Show that: (i) $\triangle AMC \cong \triangle BMD$
- Q8(ii): In right triangle ABC, right angled at $C$, $M$ is the mid-point of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM = CM$. Point $D$ is joined to point $B$ (see Fig. 7.23). Show that: (ii) $\angle DBC$ is a right angle.
- Q8(iii): In right triangle ABC, right angled at $C$, $M$ is the mid-point of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM = CM$. Point $D$ is joined to point $B$ (see Fig. 7.23). Show that: (iii) $\triangle DBC \cong \triangle ACB$
- Q8(iv): In right triangle ABC, right angled at $C$, $M$ is the mid-point of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM = CM$. Point $D$ is joined to point $B$ (see Fig. 7.23). Show that: (iv) $CM = \frac{1}{2} AB$
Other Exercises for CBSE - Class 9 Mathematics Triangles
Chapters in CBSE - Class 9 Mathematics
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Indiranagar, Bangalore 
5 yrs of Exp 
4 students 
553 hrs of classes completed 
500 per hour 
Online Classes I am a language enthusiast with an affinity for the French language. For almost a year I have been working with a Hyderabad based education company, as their online French trainer. I cater to school students, working professionals, and other French learning aspirants on a one on one basis, over an online platform, Zoom. I prepare students for the DELF beginner (A1) and elementary(A2) level exams. I also guide students with their school curriculum. I always try my best to keep the classes interactive and fun.
Super Tutor Sector 5, Gurgaon 14 yrs of Exp 14 students 2354 hrs of classes completed 400 per hour Online Classes 
Tutor's home Mathematics is not just a subject—it is the backbone of science and the foundation for logical thinking and problem-solving. With over eleven years of teaching experience and a B.Tech degree, I am dedicated to nurturing students' mathematical abilities while instilling confidence and a love for learning. Recognitions & Achievements ✅ UrbanPro Excellency Awards – Top 10 in Class 10 Tuition (2024) & Top 100 Tutors for 2024 ✅ Teacher Innovation Award by ZIIEI (Sri Aurobindo Society) ✅ Certified by Leading Ed-Tech Platforms – UrbanPro Certification, GuruQ Certification These accolades reflect my commitment to providing high-quality education and delivering exceptional learning outcomes for my students. Why Choose Me as Your Child’s Tutor? As an educator, my primary goal is to empower students with a strong conceptual foundation while ensuring they enjoy the learning process. I believe that every child has unique strengths and learning needs, and my approach is tailored to bring out the best in them. 📌 Comprehensive Academic Support: I provide in-depth coaching in Mathematics, Science, and English for Grades V–X and specialize in Mathematics for Grades XI–XII and Undergraduate Courses. 📌 Expert in Competitive & Entrance Exam Preparation: I have successfully guided students in securing admissions to prestigious institutions and excelling in competitive exams, including: 🎯 School Entrance Exams – Sainik School, Military School, Navodaya Vidyalaya, Gurukul Entrance, Central Hindu School, Rai Sports School 🎯 Competitive & Government Job Exams – Olympiads, RIMC, NTSE, SSC MTS, CHSL, DSSSB 📌 Proven Track Record of Student Success: My students have consistently demonstrated remarkable improvement in grades and conceptual understanding. I maintain an individualized learning plan for each student, ensuring they receive personalized guidance and continuous support. Regular assessments, progress reports, and feedback sessions help track improvement and address learning gaps effectively. My Teaching Methodology & Approach I strongly believe that learning should be an engaging and interactive experience rather than rote memorization. My experiential learning approach is designed to: ✔ Encourage critical thinking and problem-solving skills ✔ Foster conceptual clarity through real-life applications ✔ Enhance creativity and independent learning To achieve this, I incorporate modern teaching tools and methodologies, such as: 🖥 Live Interactive Sessions – Engaging discussions to clarify doubts in real time 📊 PowerPoint Presentations & Animations – To visualize complex concepts easily 🖋 Whiteboard Teaching – Step-by-step breakdown of mathematical problems 📚 Comprehensive Study Materials, Worksheets & Assignments – Reinforcing learning through practice Online & Offline Classes: I conduct online classes via digital platforms and in-person sessions at my institute in Gurgaon. For students requiring additional support, I offer home tuition when feasible. Commitment to Student Success 🌟 Personalized Attention: I ensure that every student gets the individual focus they need to overcome challenges and master concepts. 🌟 Supportive Learning Environment: My classes are structured to be patient, encouraging, and motivating, allowing students to ask questions freely and develop confidence in their abilities. 🌟 Building Lifelong Skills: Beyond academics, I emphasize the development of problem-solving abilities, logical reasoning, and exam strategies that benefit students in their future careers. 🌟 Parental Involvement & Regular Feedback: I believe in open communication with parents to discuss their child’s progress, strengths, and areas for improvement. Regular performance reviews ensure that parents stay informed about their child's academic growth. Invest in Your Child’s Future Today! With a proven track record, structured teaching approach, and a passion for education, I am committed to helping students achieve excellence in Mathematics and beyond. If you are looking for a dedicated, award-winning tutor who can provide high-quality education, personalized guidance, and measurable results, I would be delighted to support your child’s learning journey. 📩 Contact me today to discuss how I can help your child excel!
Rinku sir is good with science and maths for IB. My daughter is in MYP4. She has improved on the basics and numericals.
Top Tutor Madhurawada, Visakhapatnam 20 yrs of Exp 17 students 4456 hrs of classes completed 300 per hour Online Classes 
Student's Home Tutor's home I mentored several students in science subject related to PHYSICS and CHEMISTRY for class 9 for the last 24 years for CBSE/ICSE/STATE/NIOS.
As per the first class about the direction and displacement understood the concept explained..Sir is very supportive in explaining and make me understand with examples.
Top Tutor Kurichimuttom, Kozhencherry 8 yrs of Exp 1 student 379 hrs of classes completed 350 per hour Online Classes Working as a Tution teacher for the last 8 years for all classes and taught almost 150+ students.
Super Tutor New Friends Colony, Delhi 10 yrs of Exp 38 students 783 hrs of classes completed 500 per hour Online Classes Strong Maths performance starts with understanding, not memorising. I help Class 9 and 10 CBSE students build conceptual clarity, problem-solving skills, and exam confidence. My classes focus on: ✓ Complete syllabus support ✓ Concept clarification ✓ NCERT-based practice ✓ Application-based questions ✓ Case-study questions ✓ Higher-order thinking questions ✓ Previous-year questions ✓ Regular tests ✓ Error analysis ✓ Board examination strategy I particularly help students who: • Understand the chapter but struggle with questions • Make repeated mistakes • Need help with application-based problems • Want to improve their examination performance • Need structured preparation for Class 10 Boards My approach: Concept → Practice → Application → Test → Analyse → Improve Classes are personalised according to the student's current level and goals. Book an introductory session to discuss your child's Maths requirements.
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