{"id":9320,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9320"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"congruence-in-lines-and-angles","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/congruence-in-lines-and-angles\/","title":{"rendered":"Congruence In Lines And Angles"},"content":{"rendered":"

Unit: <\/strong>Congruency<\/strong><\/h2>\n

Chapter: <\/strong>Congruence in Lines and Angles<\/strong><\/h3>\n

Reference: – Basic Angle Relationships, Parallel Lines and Transversals, Perpendicular Lines and Angle Congruence, Triangle Angle Properties, Angle Congruence Theorems & Proofs, Applications & Problem-Solving<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • Geometry: – Lines & Angles<\/li>\n
  • Basic Angle Relationships & Parallel Lines and Transversals<\/li>\n
  • Perpendicular Lines and Angle Congruence<\/li>\n
  • Triangle Angle Properties & Angle Congruence Theorems & Proofs<\/li>\n<\/ul>\n

    Geometry: Lines and Angles<\/strong><\/p>\n

    Theorems:<\/strong><\/p>\n

    Axiom 1<\/strong>– If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.<\/u><\/em><\/p>\n

                           <\/em>\"\"<\/p>\n

    As we can see that PR<\/strong> is a straight line, thus angle formed on a straight line is 180°.<\/p>\n

    Therefore, ∠<\/strong>A + <\/strong>∠<\/strong>B = 180°, <\/strong>thus the sum of adjacent angles is equal to 180°.<\/p>\n

    Also, when the sum of two adjacent angles is 180°, then they are called a linear pair of angles.<\/strong><\/p>\n

    Axiom 2- <\/strong>If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.<\/u><\/em><\/p>\n

      \n
    • The two axioms above together are called the Linear Pair Axiom.<\/strong><\/li>\n<\/ul>\n

      Theorems related to Lines and Angles<\/u><\/strong><\/p>\n

      Theorem 1<\/strong>– If two lines intersect each other, then the vertically opposite angles are equal.<\/u><\/em><\/p>\n

                             <\/strong>\"\"<\/p>\n

      Proof- <\/strong>∠<\/strong>A & <\/strong>∠<\/strong>C <\/strong>and ∠<\/strong>B & <\/strong>∠<\/strong>D<\/strong> are vertically opposite angles. <\/strong><\/p>\n

      As we can see, MN is a straight line and ∠A & ∠B are adjacent angles on it,<\/p>\n

      ∠A + ∠B = 180°     …. (a) (Axiom 1)<\/strong><\/p>\n

      Similarly, <\/strong>PQ is also a straight line and ∠A & ∠D are adjacent angles on it, so<\/p>\n

      ∠A + ∠D = 180°     …. (b) (Axiom 1)<\/strong><\/p>\n

      Equating (a) & (b), we can say that ∠B = <\/strong>∠<\/strong>D.<\/strong><\/p>\n

      Similarly, we can proof this theorem for ∠<\/strong>A & <\/strong>∠<\/strong>C.<\/strong><\/p>\n

      Transversal Line<\/u><\/strong><\/p>\n

      A line which intersects two or more lines at distinct points is called a transversal line.<\/strong><\/p>\n

                            \"\"<\/p>\n

      Line R<\/strong> intersects lines P<\/strong> and Q<\/strong> at points X<\/strong> and Y<\/strong> respectively. Therefore, line R<\/strong> is a transversal for lines P<\/strong> and Q<\/strong>.<\/p>\n

       <\/p>\n

       <\/p>\n

      Here, we can observe that four angles are formed at each of the points X and Y.<\/p>\n

      Let us name them, ∠1, ∠2, ∠3 … ∠7 & ∠8.<\/p>\n

      Nomenclature of angles related to transversal line.<\/p>\n

                            \"\"<\/p>\n

        \n
      • ∠ 1, ∠ 2, ∠ 7 and ∠ 8 are called exterior angles<\/strong> and ∠ 3, ∠ 4, ∠ 5 and ∠ 6 are called interior angles<\/strong>.<\/li>\n
      • Corresponding angles<\/strong>–\n
          \n
        • ∠ 1 and ∠ 5<\/li>\n
        • ∠ 2 and ∠ 6<\/li>\n
        • ∠ 4 and ∠ 8<\/li>\n
        • ∠ 3 and ∠ 7<\/li>\n<\/ul>\n<\/li>\n
        • Alternate interior angles-<\/strong>\n
            \n
          • ∠ 4 and ∠ 5<\/li>\n
          • ∠ 3 and ∠ 6<\/li>\n<\/ul>\n<\/li>\n
          • Alternate exterior angles-<\/strong>\n
              \n
            • ∠ 1 and ∠ 8<\/li>\n
            • ∠ 2 and ∠ 7<\/li>\n<\/ul>\n<\/li>\n
            • Consecutive interior angles-<\/strong>\n
                \n
              • ∠ 4 and ∠ 6<\/li>\n
              • ∠ 3 and ∠ 5<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                Relationship between angles<\/u><\/strong><\/p>\n

                Axiom 3<\/strong>– If a transversal line intersects two parallel lines, then each pair of corresponding angles is equal.<\/u><\/em><\/p>\n

                Therefore, <\/p>\n

                  \n
                • \n
                    \n
                  • ∠<\/strong> 1 = <\/strong>∠<\/strong> 5<\/strong><\/li>\n
                  • ∠<\/strong> 2 = <\/strong>∠<\/strong> 6<\/strong><\/li>\n
                  • ∠<\/strong> 4 = <\/strong>∠<\/strong> 8<\/strong><\/li>\n
                  • ∠<\/strong> 3 = <\/strong>∠<\/strong> <\/strong>7 <\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                    Also,<\/p>\n

                    Axiom 4-<\/strong> If a transversal line intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.<\/u><\/em><\/p>\n

                    Theorem 2- <\/strong>If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.<\/u><\/em><\/p>\n

                    Proof-<\/strong> <\/u><\/em>As we know ∠<\/strong>QXY = <\/strong>∠<\/strong>AXP<\/strong>……. (Vertically Opposite angles)<\/p>\n

                    Also,            ∠<\/strong>AXP = <\/strong>∠<\/strong>XYR<\/strong>….. (Axiom 3, Corresponding angles)<\/p>\n

                    Equating both, we can conclude that,<\/p>\n

                    ∠<\/strong>QXY = <\/strong>∠<\/strong>XYR<\/strong><\/p>\n

                                    \"\"<\/p>\n

                    Similarly, we can prove this for,<\/p>\n

                    ∠<\/strong>PXY = <\/strong>∠<\/strong>XYS<\/strong><\/p>\n

                    Converse, of Theorem 2 is also true.<\/p>\n

                    Therefore,<\/p>\n

                    Theorem 3-<\/strong> If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines is parallel.<\/u><\/em> <\/strong><\/p>\n

                    Theorem 4-<\/strong> If a transversal intersects two parallel lines, then the consecutive interior angles are supplementary.<\/u><\/em><\/p>\n

                    So, <\/strong>∠<\/strong>QXY + <\/strong>∠<\/strong>XYS = 180°     <\/strong>and ∠<\/strong>PXY + <\/strong>∠<\/strong>XYR = 180°<\/strong><\/p>\n

                    Converse, of Theorem 4 is also true, so,<\/p>\n

                     Theorem 5-<\/strong> If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines is parallel.<\/u><\/em>                             <\/p>\n

                    Lines Parallel to Same Lines<\/strong><\/p>\n

                             \"\"<\/p>\n

                    Given, Line A\u2225<\/em>\"\" B and A\u2225<\/em>\"\" C, where line P is the transversal line.<\/p>\n

                     <\/p>\n

                    So as, A\u2225<\/em> B, ∠1= ∠2….. (Corresponding angles)<\/p>\n

                    Also, A\u2225<\/em> C, ∠1= ∠3….. (Corresponding angles)<\/p>\n

                    Thus, we can say that ∠<\/strong>2 = <\/strong>∠<\/strong>3<\/strong><\/p>\n

                    Now as ∠2 = ∠3, we can say that line B<\/strong> <\/strong>\u2225<\/em>  C.<\/strong><\/p>\n

                    Theorem 6<\/strong>– Lines which are parallel to the same line are parallel to each other.<\/u><\/em>
                    \n <\/p>\n

                      \n
                    • Congruent Angles<\/strong> – Two angles are congruent if they have the same measure.<\/li>\n
                    • Angle Addition Postulate<\/strong> – If a point lies inside an angle, the sum of the two smaller angles equals the measure of the larger angle.<\/li>\n
                    • Complementary Angles<\/strong> – Two angles are complementary if the sum of their measures is 90°.<\/li>\n
                    • Supplementary Angles<\/strong> – Two angles are supplementary if the sum of their measures is 180°.<\/li>\n
                    • Vertical Angles<\/strong> – When two lines intersect, the opposite (vertical) angles formed are always congruent.<\/li>\n
                    • Parallel Lines and Transversals<\/strong> – When a transversal intersects two parallel lines, special angle relationships are created, including corresponding, alternate interior, alternate exterior, and consecutive interior angles.<\/li>\n
                    • Perpendicular Lines<\/strong> – Two lines are perpendicular if they intersect at a 90° angle.<\/li>\n
                    • Perpendicular Bisector Theorem<\/strong> – A point on the perpendicular bisector of a segment is equidistant from the segment’s endpoints.<\/li>\n
                    • Right Angles Congruence Theorem<\/strong> – All right angles are congruent, meaning they have the same measure of 90°.<\/li>\n
                    • Triangle Sum Theorem<\/strong> – The sum of the interior angles of any triangle is always 180°.<\/li>\n
                    • Exterior Angle Theorem<\/strong> – The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.<\/li>\n
                    • Congruent Supplements Theorem<\/strong> – If two angles are supplementary to the same angle, then they are congruent.<\/li>\n
                    • Congruent Complements Theorem<\/strong> – If two angles are complementary to the same angle, then they are congruent.<\/li>\n
                    • Geometric Proofs<\/strong> – A logical sequence of statements and reasons used to justify geometric relationships and congruence.<\/li>\n<\/ul>\n

                       <\/p>\n","protected":false},"excerpt":{"rendered":"

                      Unit: Congruency Chapter: Congruence in Lines and Angles Reference: – Basic Angle Relationships, Parallel Lines and Transversals, Perpendicular Lines and Angle Congruence, Triangle Angle Properties, Angle Congruence Theorems & Proofs, Applications & Problem-Solving After studying this chapter, you should be able to understand: Geometry: – Lines & Angles Basic Angle Relationships & Parallel Lines and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[632],"tags":[],"class_list":["post-9320","post","type-post","status-publish","format-standard","hentry","category-high-school-geometry"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9320","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9320"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9320\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}