{"id":222,"date":"2022-04-13T08:45:15","date_gmt":"2022-04-13T08:45:15","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=222"},"modified":"2025-12-19T06:17:49","modified_gmt":"2025-12-19T06:17:49","slug":"m-3-construct-a-congruent-angle","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/m-3-construct-a-congruent-angle\/","title":{"rendered":"M.3 Construct a congruent angle"},"content":{"rendered":"\n

Construct a congruent angle<\/strong><\/h2>\n\n\n\n

key notes :<\/p>\n\n\n\n

\"https:\/\/i.ytimg.com\/vi\/c2x0NATrBIY\/maxresdefault.jpg?utm_source=chatgpt.com\"<\/figure>\n\n\n\n
\"https:\/\/study.com\/cimages\/videopreview\/videopreview-full\/eq5w7siub2.jpg?utm_source=chatgpt.com\"\/<\/figure>\n\n\n\n
\"https:\/\/cdn.prod.website-files.com\/6634a8f8dd9b2a63c9e6be83\/669fbf196785d1203e484493_9781119181552-fg2008.gif?utm_source=chatgpt.com\"\/<\/figure>\n\n\n\n

\ud83d\udd39 Meaning<\/h3>\n\n\n\n
    \n
  • Congruent angles<\/strong> are angles that have the same measure<\/strong>.<\/li>\n\n\n\n
  • Constructing a congruent angle means copying an angle exactly<\/strong> using geometric tools.<\/li>\n<\/ul>\n\n\n\n

    \ud83d\udd39 Tools Required<\/h3>\n\n\n\n
      \n
    • Compass<\/strong><\/li>\n\n\n\n
    • Ruler \/ Straightedge<\/strong><\/li>\n\n\n\n
    • Pencil<\/strong><\/li>\n<\/ul>\n\n\n\n

      \ud83d\udd39 Given<\/h3>\n\n\n\n
        \n
      • A given angle \u2220ABC<\/li>\n\n\n\n
      • A point (say P<\/strong>) where the congruent angle must be constructed<\/li>\n<\/ul>\n\n\n\n

        \ud83d\udd39 Steps to Construct a Congruent Angle<\/h3>\n\n\n\n
          \n
        1. Draw a ray<\/strong> starting from point P<\/strong> (this will be one arm of the new angle).<\/li>\n\n\n\n
        2. Place the compass at the vertex of the given angle<\/strong> and draw an arc<\/strong> that cuts both arms of the angle.<\/li>\n\n\n\n
        3. Without changing the compass width, draw the same arc<\/strong> from point P<\/strong> on the new ray.<\/li>\n\n\n\n
        4. Measure the distance between the two arc intersection points<\/strong> on the given angle using the compass.<\/li>\n\n\n\n
        5. With the same distance, mark a point on the new arc drawn from P<\/strong>.<\/li>\n\n\n\n
        6. Join point P to the marked point<\/strong>.
          \ud83d\udc49 The angle formed is congruent<\/strong> to the given angle.<\/li>\n<\/ol>\n\n\n\n

          \ud83d\udd39 Key Properties<\/h3>\n\n\n\n
            \n
          • The measure of both angles is equal<\/strong>.<\/li>\n\n\n\n
          • The construction uses only compass and ruler<\/strong> (no protractor).<\/li>\n\n\n\n
          • Lengths and arcs used are kept unchanged<\/strong>.<\/li>\n<\/ul>\n\n\n\n

            \ud83d\udd39 Important Notes<\/h3>\n\n\n\n
              \n
            • Do not change the compass opening<\/strong> during the steps.<\/li>\n\n\n\n
            • Accuracy depends on careful marking of arcs<\/strong>.<\/li>\n\n\n\n
            • This method works for acute, obtuse, and right angles<\/strong>.<\/li>\n<\/ul>\n\n\n\n

              \ud83d\udd39 Applications<\/h3>\n\n\n\n
                \n
              • Used in geometric constructions<\/strong><\/li>\n\n\n\n
              • Helps in triangle constructions<\/strong><\/li>\n\n\n\n
              • Important for proofs involving congruence<\/strong><\/li>\n<\/ul>\n\n\n\n

                \ud83d\udd39 Example Statement<\/h3>\n\n\n\n
                  \n
                • If \u2220ABC = 45\u00b0, then the constructed angle \u2220PQR = 45\u00b0<\/strong>, hence
                  \u2220ABC \u2245 \u2220PQR<\/strong><\/li>\n<\/ul>\n\n\n\n

                  Learn with an example<\/strong><\/p>\n\n\n\n

                  \n
                  \n

                  \ud83c\udf00 The diagram below shows a nearly completed construction of an angle congruent to \u2220A with vertex D and side DE.<\/strong><\/p>\n\n\n\n

                  \u2a00D was drawn with radius AB<\/em>.<\/p>\n\n\n\n

                  \u2a00G was drawn with radius BF<\/em>.<\/p>\n\n\n\n

                  Complete the construction.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div><\/div><\/div>\n\n\n\n

                  Part of the construction was done for you. Here are the steps to create this part of the construction.<\/p>\n\n\n\n

                  Start with the objects in the diagram below.<\/strong><\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Draw a circle with radius AB centred at A.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Mark the point where \u2a00A and AC intersect. Call it F.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Since B and F are both on \u2a00A, AB<\/em>=AF<\/em>.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Draw a circle with radius AB centred at D.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Mark the point where \u2a00D and DE intersect. Call it G.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Since \u2a00D has radius AB<\/em>, DG<\/em>=AB<\/em>.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Draw a circle with radius BF centred at G.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Mark a point where \u2a00D and \u2a00G intersect. Call it H.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Since \u2a00G has radius BF<\/em>, GH<\/em>=BF<\/em>.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Since \u2a00D has radius AB<\/em>, DH<\/em>=AB<\/em>=AF<\/em>.<\/p>\n\n\n

                  \n
                  \"\"<\/figure><\/div>\n\n\n

                  Complete the construction.<\/p>\n\n\n\n

                  To complete the construction of an angle congruent to \u2220A with vertex D and side DE, carry out the following step:<\/p>\n\n\n\n