{"id":10096,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10096"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","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":"
\n
\n

KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

\n

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\u00b0.<\/u><\/em><\/p>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/p>\n

    \"\"<\/p>\n

    Source: Kapdec.com<\/p>\n<\/div>\n

    As we can see that PR<\/strong> is a straight line, thus angle formed on a straight line is 180\u00b0.<\/p>\n

    Therefore, \u2220<\/strong>A + <\/strong>\u2220<\/strong>B = 180\u00b0, <\/strong>thus the sum of adjacent angles is equal to 180\u00b0.<\/p>\n

    Also, when the sum of two adjacent angles is 180\u00b0, then they are called a linear pair of angles.<\/strong><\/p>\n

    Axiom 2- <\/strong>If the sum of two adjacent angles is 180\u00b0, 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

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Proof- <\/strong>\u2220<\/strong>A & <\/strong>\u2220<\/strong>C <\/strong>and \u2220<\/strong>B & <\/strong>\u2220<\/strong>D<\/strong> are vertically opposite angles. <\/strong><\/p>\n

      As we can see, MN is a straight line and \u2220A & \u2220B are adjacent angles on it,<\/p>\n

      \u2220A + \u2220B = 180\u00b0\u00a0\u00a0\u00a0\u00a0 …. (a) (Axiom 1)<\/strong><\/p>\n

      Similarly, <\/strong>PQ is also a straight line and \u2220A & \u2220D are adjacent angles on it, so<\/p>\n

      \u2220A + \u2220D = 180\u00b0\u00a0\u00a0\u00a0\u00a0 …. (b) (Axiom 1)<\/strong><\/p>\n

      Equating (a) & (b), we can say that \u2220B = <\/strong>\u2220<\/strong>D.<\/strong><\/p>\n

      Similarly, we can proof this theorem for \u2220<\/strong>A & <\/strong>\u2220<\/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

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\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

      \u00a0<\/p>\n

      \u00a0<\/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, \u22201, \u22202, \u22203 … \u22207 & \u22208.<\/p>\n

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

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

        \n
      • \u2220 1, \u2220 2, \u2220 7 and \u2220 8 are called exterior angles<\/strong> and \u2220 3, \u2220 4, \u2220 5 and \u2220 6 are called interior angles<\/strong>.<\/li>\n
      • Corresponding angles<\/strong>–\n
          \n
        • \u2220 1 and \u2220 5<\/li>\n
        • \u2220 2 and \u2220 6<\/li>\n
        • \u2220 4 and \u2220 8<\/li>\n
        • \u2220 3 and \u2220 7<\/li>\n<\/ul>\n<\/li>\n
        • Alternate interior angles-<\/strong>\n
            \n
          • \u2220 4 and \u2220 5<\/li>\n
          • \u2220 3 and \u2220 6<\/li>\n<\/ul>\n<\/li>\n
          • Alternate exterior angles-<\/strong>\n
              \n
            • \u2220 1 and \u2220 8<\/li>\n
            • \u2220 2 and \u2220 7<\/li>\n<\/ul>\n<\/li>\n
            • Consecutive interior angles-<\/strong>\n
                \n
              • \u2220 4 and \u2220 6<\/li>\n
              • \u2220 3 and \u2220 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,\u00a0<\/p>\n

                  \n
                • \n
                    \n
                  • \u2220<\/strong> 1 = <\/strong>\u2220<\/strong> 5<\/strong><\/li>\n
                  • \u2220<\/strong> 2 = <\/strong>\u2220<\/strong> 6<\/strong><\/li>\n
                  • \u2220<\/strong> 4 = <\/strong>\u2220<\/strong> 8<\/strong><\/li>\n
                  • \u2220<\/strong> 3 = <\/strong>\u2220<\/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 \u2220<\/strong>QXY = <\/strong>\u2220<\/strong>AXP<\/strong>……. (Vertically Opposite angles)<\/p>\n

                    Also,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2220<\/strong>AXP = <\/strong>\u2220<\/strong>XYR<\/strong>….. (Axiom 3, Corresponding angles)<\/p>\n

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

                    \u2220<\/strong>QXY = <\/strong>\u2220<\/strong>XYR<\/strong><\/p>\n

                    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/p>\n

                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

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

                    \u2220<\/strong>PXY = <\/strong>\u2220<\/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>\u2220<\/strong>QXY + <\/strong>\u2220<\/strong>XYS = 180\u00b0\u00a0\u00a0\u00a0\u00a0 <\/strong>and \u2220<\/strong>PXY + <\/strong>\u2220<\/strong>XYR = 180\u00b0<\/strong><\/p>\n

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

                    \u00a0Theorem 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>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

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

                    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/p>\n

                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    Given, Line A\u2225<\/em><\/p>\n

                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    B and A\u2225<\/em><\/p>\n

                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    C, where line P is the transversal line.<\/p>\n

                    \u00a0<\/p>\n

                    So as, A\u2225<\/em> B, \u22201= \u22202….. (Corresponding angles)<\/p>\n

                    Also, A\u2225<\/em> C, \u22201= \u22203….. (Corresponding angles)<\/p>\n

                    Thus, we can say that \u2220<\/strong>2 = <\/strong>\u2220<\/strong>3<\/strong><\/p>\n

                    Now as \u22202 = \u22203, we can say that line B<\/strong> <\/strong>\u2225<\/em> \u00a0C.<\/strong><\/p>\n

                    Theorem 6<\/strong>– Lines which are parallel to the same line are parallel to each other.<\/u><\/em>
                    \n\u00a0<\/p>\n

                      \n
                    • Congruent Angles<\/strong> \u2013 Two angles are congruent if they have the same measure.<\/li>\n
                    • Angle Addition Postulate<\/strong> \u2013 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> \u2013 Two angles are complementary if the sum of their measures is 90\u00b0.<\/li>\n
                    • Supplementary Angles<\/strong> \u2013 Two angles are supplementary if the sum of their measures is 180\u00b0.<\/li>\n
                    • Vertical Angles<\/strong> \u2013 When two lines intersect, the opposite (vertical) angles formed are always congruent.<\/li>\n
                    • Parallel Lines and Transversals<\/strong> \u2013 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> \u2013 Two lines are perpendicular if they intersect at a 90\u00b0 angle.<\/li>\n
                    • Perpendicular Bisector Theorem<\/strong> \u2013 A point on the perpendicular bisector of a segment is equidistant from the segment\u2019s endpoints.<\/li>\n
                    • Right Angles Congruence Theorem<\/strong> \u2013 All right angles are congruent, meaning they have the same measure of 90\u00b0.<\/li>\n
                    • Triangle Sum Theorem<\/strong> \u2013 The sum of the interior angles of any triangle is always 180\u00b0.<\/li>\n
                    • Exterior Angle Theorem<\/strong> \u2013 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> \u2013 If two angles are supplementary to the same angle, then they are congruent.<\/li>\n
                    • Congruent Complements Theorem<\/strong> \u2013 If two angles are complementary to the same angle, then they are congruent.<\/li>\n
                    • Geometric Proofs<\/strong> \u2013 A logical sequence of statements and reasons used to justify geometric relationships and congruence.<\/li>\n<\/ul>\n

                      \u00a0<\/p>\n

                      <\/p>\n